
Class _J^i_:_: 



GopghtN^ 



COP^-RIGHT DEPOSIT. 



PRACTICE IN THE CASE OF 
SCHOOL CHILDREN 



BY 

THOMAS JOSEPH KIRBY, Ph.D. 

PROFESSOR OP ELEMENTARY EDUCATION AT THE UNIVERSITY OF PITTSBURGH 



TEACHERS COLLEGE, COLUMBIA UNIVERSITY 
CONTRIBUTIONS TO EDUCATION, NO. 58 



PUBLISHED BY 

^tntijjtrB CHolUge, Qlolttmbia 3lntttprHttg 

NEW YORK CITY 

1913 



/!/) V xp^ 



y 



vP^« 



Copyright, l})i;{, by Thomas Joseph Kirby 



4U i 



ACKNOWLEDGMENTS 

No acknowledgments I can make adequately show my obliga- 
tions to Dr. Edward L. Thorndike for assistance in connection 
with this study. To him I am indebted for suggesting and out- 
lining the study, for guidance during its progress, for assistance 
in computing and interpreting the results, and for reading and 
revising the manuscript. 

To Dr. J. McKeen Cattell and Dr. R. S. Woodworth I am 
indebted for valuable suggestions received during reports on the 
study in their seminar. To Dr. C. L. Robbins 1 am under obli- 
gations for careful checking of the statistical results, and to Dr. 
H. W. Reddick for a careful reading of the manuscript. To the 
children, teachers, principals, and superintendent of the Chil- 
dren's Aid Society of New York City, I am indebted for cheer- 
ful cooperation during the progress of the practice. To my 
sister, Miss Nelle Kirby, I am indebted for conducting the 
practice of one class reported in this study. 

T. J. K. 



CONTENTS 

Chapter Page 

I The Administration of the Exi-eiuments i 

Children Participating i 

Conductors of the Practice 2 

Material Used 2 

Plan of the Practice 3 

Method of Conducting the Practice 6 

Method of Scoring 9 

Efforts for Uniformity 10 

II Improvement in the Group as a Whole 12 

Addition 12 

Initial Ability i5 

Accuracy 16 

Gross Gain in Numl)er of Problems Correctly Added 18 

Relative Gain in Addition from Sixty Minutes of Practice.. 20 

Gross Gain in Accuracy in Addition 22 

Summary 24 

Division 24 

Initial Ability 27 

Accuracy in Division 28 

Relative Gain in Division 31 

Gross Gain in Accuracy 32 

Summary 34 

The Value of the Practice Experiment as a Method of Teaching 35 

Factors Contributing to the Improvement 37 

III The Effect of the Distribution and Length of Work Period 

UPON the Rate of Learning 45 

Plan of Practice 45 

Addition 47 

Initial Ability of the Groups 47 

Gross Gain 49 

Percentile Gain Si 

Gain in Accuracy 52 

Summary 54 

Division 55 

Initial Ability 55 

Gross Gain 59 

Percentile Gain 60 

Gain in Accuracy 60 

Summary oO 

Improvement in Relation to Initial Ability 68 

General Summary 69 



vi Contents 

Chapter Page 

IV The Permanence of the Practice Effect 71 

Permanence of Association Normally Used in School Work... 72 

Addition 72 

Gross Gain 73 

Gain in Accuracy 73 

Division 75 

Gross Gain 75 

Gain in Accuracy 76 

Summary jy 

Permanence of Associations Through Summer Vacation 78 

Addition 78 

Gross Loss 78 

Loss Per Cent 78 

Loss in Accuracy 79 

Division 79 

Gross Loss 80 

Loss Per Cent 81 

Loss in Accuracy 8i 

Permanence in Terms of Advance Over the Initial Practice- 
Period of the Original Experiment 81 

Addition 81 

Gross Gain 81 

Change in Accuracy 82 

Division 83 

Gross Gain 83 

Loss in Accuracy 84 

Permanence of Associations in Addition and Division during 
Vacation as Shown by the Amount of Practice Required to 

Restore these Associations to their Previous Efficiency 86 

Addition 86 

Time of Practice 86 

Gross Gain 87 

Accuracy 87 

Division 89 

Time of Practice 89 

Gross Gain 90 

Accuracy 90 

Appendix I. The Use of the Method of the Practice Experiment in 

Teaching Handwriting and Spelling 93 

Appendix II. Sample of Addition and Division Sheets Used 97 



PRACTICE IN THE CASE OF SCHOOL CHILDREN 

CHAPTER I 
THE ADMINISTRATION OF THE EXPERIMENTS 

Children Participating 

The practice on which this study is based was conducted dur- 
ing the years 191 1 and 1912 in the schools of the Children's Aid 
Society in the third- and fourth-year classes as a part of the 
regular grade work in arithmetic. In all, about 1350 children 
took part. It was the intention that the study should be con- 
ducted under school conditions normal for children, teachers, 
and supervisor, in order to meet, as far as possible, a current 
criticism that results obtained from studies with small groups 
of persons under laboratory conditions, are not applicable in 
school conditions ; and also in order to establish greater confi- 
dence in the validity of the results of the study and insure the 
applicability of its results to actual school-room problems. It is 
true that one meets a greater number of factors under such 
conditions that can not be controlled than in an experiment con- 
ducted under laboratory conditions, but they are the factors 
that enter into all school-room work and help to detemiine both its 
quality and quantity. There was approximately the same simi- 
larity of ability in the classes that took the practice that one 
ordinarily finds in a school system where a common course of 
study is followed and where all the teachers are under similar 
supervision, but where there is diversity of nationality and ex- 
treme individual variations among the children. 

Two different experiments were conducted, one in addition, 
the other in division. The addition was given to classes in the 
fourth year of the elementary school ; the division, to classes 
in the last half of the third year and the first half of the fourth 
year. These classes had done the work outlined in arithmetic 
in the New York City course of study. Hence the children had 
been taught the facts and processes involved in the experiment. 

I 



2 Practice in the Case of School Children 

Their exact proficiency in these functions at the beginning of 
tlic experiment will be defined more accurately later by means 
of tiieir records in the first practice-period. 

Conductors of the Practice 

Thirty-nine classes took part in the practice. The writer con- 
ducted the entire practice in thirty-four of these classes. In 
four other classes (VT, X, XXV, and XXVI) he had charge 
of the initial and final practice-periods and of some others, but 
some of the intervening practice-periods were in charge of the 
principal of the school or the teacher of the class, who followed 
the same plan as the writer. They could do this because they 
were present during all the previous i)enods of practice and 
had been asked to note the exact method of procedure in order 
that they might be able to duplicate it exactly when the writer 
could not be present. Their interest and fine spirit of coopera- 
tion insured the most careful observance of all the requirements 
of the experiment. Since the writer usually gave the practice- 
period preceding and following one given by them, it was pos- 
sible to note any variations that might occur in results. No 
greater variations were noticeable in the performance of these 
classes at such times from one day to the next than when the 
writer conducted the tests continuously. In the case of class XX, 
the complete i)ractice was conducted by the class-room teacher 
in a school outside of New York City. All the details of the 
experiment were carefully observed. The results in this class 
conformed so closely to those in other classes that it seemed 
perfectly justifiable to use them in order to make the number 
of classes in the difl^erent groups equal. Anyone who wishes 
to do so, will find that the results are practically uninfluenced by 
omitting these five classes from the totals. 

Material Used 

In the addition practice, the Thorndike 'Addition Sheets* 
(which consist of seven different sheets, each containing 48 
columns of one-place numbers, each column containing ten ad- 
dends, I's and o's omitted, so arranged that any successive five 
of the columns are of nearly equal difficulty) were used. By 
using the seven diflFerent sheets, the chance of remembering 



The Administration of the Experiments 3 

answers to any column because of its position is reduced to a 
minimum. These sheets are printed in large clear type on un- 
glazed paper which obviates eye-strain with its attending fatigue, 
and gives the best possible conditions for rapid, concentrated, 
accurate work. A sample of one of these sheets may be found 
on page 97. 

For the practice in division, sheets were devised by the writer 
on the " Remainder Division Table " plan worked out by Thorn- 
dike in his " Exercises in Arithmetic," Nos. 2 and 3. These 
division combinations include the entire series from " 20 ^ — 3s 
and — remainder" up to "89 = — 9s and — remainder," thus 
involving not only the combinations which are the inverse of 
the multiplication tables through 9 times 9, but also the division 
of the intervening series of numbers which give a remainder in 
the answer. No dividends smaller than twenty were used be- 
cause they involve associations which are usually well established 
in children's minds and so need no further drill. A sample of 
these sheets may be seen on page 98. Three sheets of these 
combinations were arranged in such a way as to make any group 
of eight or ten successive combinations practically equal in diffi- 
culty to any other such group. This arrangement was the only 
means of obviating the inequality of difficulty in individual com- 
binations. An examination of these sheets will convince one 
that the difficulty involved in any continuous group of eight or 
ten combinations is, for all practical purposes, the same as that 
for any other similar group, even though the individual com- 
binations may differ widely in degree of difficulty. The records 
made by classes on these tests are further evidence of the equality 
of different groups of eight or ten successive combinations. 

Plan of the Practice 

In the addition experiment each class practiced adding these 
columns for 75 minutes ; but this time was distributed differently 
for different classes. Each class had an initial practice-period 
of fifteen minutes. These were identical in character with the 
intervening practice-periods ; but besides serving as practice- 
periods, they served as measures of ability at the beginning and 
end of practice. From them the change in ability for each indi- 
vidual was measured. The intervening practice of forty-five 



4 Practice in the Case of School Children 

minutes between the initial practice-period and the final practice- 
period was broken up for different groups of classes in four 
different ways. For Group I it was divided into two practice- 
periods of 22^ minutes each ; for Group II, into three practice- 
periods of 15 minutes each; for Group III, into eight practice- 
periods, seven of six minutes each and one of three minutes ; 
for Group IV into twenty-two practice-periods, twenty-one of 
two minutes each and one of three minutes. The following 
statement of this plan in tabular form adds to clearness : 



Guoui's 


Initial rioiuou 


Inti' 


;i{V10NING 45 MlNUTKS 


Final Piouiod 


I 


15 min. 


•) 


22 i min. 


15 min. 


II 


15 min. 


ii 


15 min. 


15 min. 


111 


15 min. 


7 


G min. and 1 .'} min. 


15 min. 


IV 


15 min. 


21 


2 min. and 1 ;i min. 


15 min. 



It is of special importance for an understanding of the entire 
discussion to get clearly in mind that all the classes practiced 
for the same aggregate amount of time. It is further to be 
carefully noted that for all classes the initial period and final 
period of practice were of equal length. Since all four groups 
had the same initial and final practice-periods, but had the inter- 
vening forty-five minutes of practice distributed in different 
ways, any difference in the results for the groups will be a 
consequence of distributing a given amount of time in periods 
of dilTcrcnt length.' 

In the division experiment the aggregate time of practice was 
sixty minutes for each class. This was divided into periods of 
different lengths for different groups of classes. Each class had 
an initial practice-period of ten minutes and a final period of ten 
minutes ; and the intervening practice of 40 minutes was divided 
in three different ways. In the division, as in the addition, the 
initial and final practice-periods were identical in character with 
the intervening practice-periods ; but besides serving as practice- 
periods, they served as measures of ability at the beginning and 
end of practice from which any change in ability was measured. 
The first group had two intervening practice-periods of twenty 
minutes each ; a second group had four intervening practice- 
periods of ten minutes each; a third group had twenty inter- 
vening practice-periods of two minutes each. The following 
tabular statement helps to kcc]) the plan for each group in mind : 

^ Except iov a factor to ho noted on page 62 ff. 



The Administration of the Experiments 



tours 


Initial Piomou 


iNTEllVENINa 40 MiNUTKH 


Final Period 


I 

II 
III 


10 min. 
10 min. 
10 min. 


2 20 min. 

4 10 min. 

20 2 min. 


10 min. 
10 min. 
10 min. 



The plan of liaving each class drill for the same aggregate 
length of time, sixty minutes, makes it i)ossiblc to measure the 
gain in all classes for this amount of drill ; while breaking the 
forty minutes intervening between the initial test and the final 
test into periods of different lengths gives the opportunity, as in 
the addition experiment, to find the comparative effect of drill- 
periods of different lengths, which aggregate a constant amount. 

As nearly as normal school conditions would i)ermit, all prac- 
tice-periods were given one each on successive school days to 
any class. So in addition, in the classes of Group I, four days 
were required to complete the experiment; of Grouj) IT, 
five days; of Group J II, ten days; and of Group IV, twenty-four 
days. In division four days were required for the classes of 
Group I ; six days for Group II ; and twenty-two days for Group 
111. In three classes it was found necessary to give a two- 
minute practice-period twice a day, one in the morning and one 
in the afternoon for the last five days in order to complete the 
experiment within a time made necessary by school conditions. 
No noticeable effect was produced by this change. The practice 
was given to any individual class as nearly at the same time of 
day as was possible. In general the time of day did not vary 
for any class more than half an hour, but sometimes it was 
necessary to give a practice-period to a class in the afternoon in- 
stead of in the morning. The teachers felt that children would 
not do so well in the afternoon, but the results showed little 
or no influence from such changes of time. Of course different 
classes did not have their practice at the same time of day. 
Some classes had most of their practice early in the morning, 
others later in the morning. Some classes had most of their 
practice early in the afternoon while others had theirs late in 
the afternoon. Some of the very best records were made late 
in the afternoon despite the general feeling on the ])art of the 
teachers that a class could not do as well late in the day. Some 
classes seemed to be affected by the loss of school work on 
vSaturday and Sunday, or Ijy the intervention of a holiday. Yet 
in other classes remarkaljle gains were made following such a 



6 Practice in the Case of School Children 

holiday, which seems to obviate the need of carefully evaluating 
any such variable factors. In fact other studies have shown 
that where interest is keen children do as effective work late in 
the day as early. Hence this factor receives no consideration. 
The fact that the interval between the initial and final practice- 
periods was larger when the intervening practice-periods were 
shorter and so gave more opportunity for practice out of school 
and for the influence of regular school training outside the 
specified practice of the experiment, conditions all the results 
of this study. TJiere were three possibilities. The practice could 
be given as it was ; or it could be given with longer intervals 
when the practice-periods were long, so as to have the final 
practice-period always come, say, 24 school-days after the initial 
practice-period ; or it could be given daily as it was, but with a 
varying interval between the next to the last and the last prac- 
tice-period so that the final practice-period would always come, 
say, 24 school-days after the initial practice-period. Each method 
has its disadvantages. The method used here gives the pupils 
])racticing in shorter periods greater chances of help from prac- 
tice outside the experiment. The second method mixes the 
effect of interval-length with the effect of practice-period-length. 
The third gives the shorter periods the advantage of less for- 
getting. In interpreting the results given in this monograph, 
the reader may allow for the greater chance of outside practice 
in the short-period groups as seems wise. 

The Method of Conducting the Practice 

The purpose of the practice was explained to the children as 
carefully as possible, and in the same way to all classes. The 
fact that much of arithmetic work both in school and out in- 
volved addition was impressed as well as the need for speed and 
accuracy in it. They were told that this work would help them 
improve their addition. The value to be derived from the prac- 
tice was indicated. In the case of addition, no preliminary 
work was given, because it could be assumed that any fourth- 
year child knew how to add columns of figures. Before the 
papers were passed out the children were shown one of the 
sheets and instructed to place their answers on the sheet which 
would be given to them, in the space provided. They were told 



The Administration of the Experiments 7 

to add as rapidly as possible, but to make as few mistakes as 
possible, as all mistakes would count against their scores. The 
need of accuracy combined with speed was impressed further 
by showing them the method of scoring, no credit save for 
each column correctly added. When asked what they were to 
do, their most common response was, "Add as fast as you can 
and don't make mistakes." They were also told how many 
minutes they were to work, but they had no means of knowing 
the time, once the signal to begin was given, until the signal 
to stop was given. 

The papers were passed out by the class teacher in her regular 
way, with the printed side down. In all the practice each child 
was supplied with more problems than he could finish in the 
allotted time. The children placed their names, age, and grades 
on the back of the sheets. Then they were instructed again to 
add as rapidly and as accurately as possible and to place their 
answers on the sheets beneath the columns. They were told 
that the signal for beginning would be " Go," and for ending 
" Pencils down." The oft-repeated injunction, " Look only at 
your own paper " was given, and was followed quite conscien- 
tiously. After a practice-period was finished and pencils were 
laid down, the children were asked about the number of problems 
they had worked. This was done because the children were 
eager to tell their own scores and to learn the scores of others. 
It acted too as an immediate reward and so as an incentive. Care 
was taken to avoid undue excitement, but an attempt was made 
to inspire the children to their best individual efforts. 

Just before beginning the second and each following practice- 
period, the exact score of each child in the preceding day's 
practice was read, both the number of columns worked and the 
number correct.- High scores with few errors were commended. 
If some score showed that perhaps this child was adding beyond 
his best speed, as indicated by the number of errors, he was 
advised, not in a fault-finding way but for the good of his own 
record, to try for greater accuracy with the hope of improving 
his score. These facts were put in concrete form for the chil- 
dren as follows : " Which gives the better score, 20 with 4 
wrong or 16 with i wrong?" Of course this warning was not 



With a very few exceptions. 



8 Practice in the Case of School Children 

always pertinent, though usually a large number of mistakes 
indicated that the child was working beyond his own norm of 
speed. At all times during the practice the value of accuracy 
was stressed ; and the undesirability of working an excessive 
number of problems with many errors was pointed out. 

Further instructions calculated to maintain effort were given 
just before beginning the second practice-period and repeated 
always at the beginning of each subsequent one. The children 
were told that their individual improvement was to be measured 
and they were shown that no matter how low or high their pres- 
ent record their final standing would be determined by the 
amount of gain made. They were shown that it was not pri- 
marily a contest among the individuals of the class, but an effort 
on the part of each one to surpass his own previous record. 
The children v/ere encouraged to compare their last record with 
their own previous records, and at times the scores were read 
to them in such a way as to indicate gains made. A hypothetical 
curve was drawn upon the board to indicate the ascent that 
would result from supposed gains, as well as to show them how 
to keep their own individual curves. 

After a few periods of practice, actual scores were used on 
the board in the construction of these curves. No doubt, had 
each child been encouraged to draw his own curve from day 
to day, this concrete showing of his gain would have resulted 
in added effort that would have justified fully the time spent. 
The desire to surpass one's own previous record was the incen- 
tive most appealed to ; still the spirit of rivalry was strong. One 
little girl expressed it well for the class by saying : '* Why, you 
just try to beat yourself," but an ambitious boy, when asked 
what record he was trying to surpass, pointed to his neighbor 
behind him and said : " I'm trying to beat him." The average 
number of problems done correctly in the preceding practice- 
period by boys and girls as separate groups was also read to the 
class before beginning the day's practice. This group com- 
parison served as an additional incentive, but would not satisfy 
the children as a substitute for their individual scores. 

In division, preliminary work was given before the initial 
practice-period was begun. The need of knowing the multiplica- 
tion and division combinations, both for school and life, was 
explained. The children were told that this work would aid 



The Administration of the Experiments 9 

them in both. Then many of the combinations were placed upon 
the blackboard to insure familiarity with the form of expression 
used in the sheets. Particular attention was given to assisting 
any child who appeared not to understand. After a seemingly 
general response was obtained in this way, in order to make 
doubly sure that each individual knew exactly what he was to 
do and how to record the result properly on the sheet, one-sixth 
of a sheet was given to each child on which to fill in the results. 
During this part of the preliminary trial very great care was 
taken by the writer and the grade teacher to see that all the 
children understood fully what was to be done. Despite this 
care the first papers of some few individuals could not be used 
because they showed clearly a failure to grasp the situation. In 
most cases these failures were from children who had recently 
entered the classes and were not yet properly classified, or who 
had not sufficient mastery of English to understand the instruc- 
tions. Yet some failed because they were backward in learning 
the process of short division. Almost without exception the chil- 
dren who failed to grasp the situation after the explanation 
mentioned above, were found to be unable to handle short 
division in any form. Later the teacher explained the matter 
further to these children. They continued in the practice, but 
their records did not enter in the final computation. In all other 
respects, the same plan was followed in the division as has been 
described in the addition, though greater stress could be placed 
on speed, inasmuch as the percentage of accuracy showed a 
gradual increase along with the absolute gain in number of com- 
binations. 

Method of Scoring 

In scoring the addition, credit was given according to columns. 
The score for any paper was found by deducting one from the 
total number of columns added for each column incorrectly 
added. If a child added 16 columns with 2 wrong, his score was 
taken as 14. Thus efficiency was measured by the rapidity and 
accuracy of getting the sums. In division a combination was 
taken as a unit, correct quotient and remainder being required 
for any credit. The score for any paper was found by deducting 
one from the total number of combinations solved for each one 



lo Practice in the Case of School Children 

incorrectly solved. If a child did 30 combinations with 2 incor- 
rect, his score was 28. One might contend that some credit 
should be given for a correct quotient with an incorrect re- 
mainder ; but after all in the use of such combinations the result 
as a whole determines efficiency. Thus ability was measured by 
the rapidity and accuracy of getting the quotients and remain- 
ders. These methods of scoring whereby speed and accuracy 
are reduced to a single variable, are entirely arbitrary, but the 
author feels reasonably certain that any system of weighting 
errors which might be used would not materially change the 
results on which conclusions are based. 

The record of no child could be used if he was absent at a 
first or last practice, inasmuch as the change in his ability could 
not be measured. But records of those who missed one or 
more intervening tests were used.'' In all, about 1350 children 
were tested. The smallest number of ])apers scored for any 
child who was present every day during the time an experiment 
was in progress was four, and the greatest number twenty-four. 
This means that in all about 15,000 papers were graded and 
their scores recorded in the process of giving the two experi- 
ments. 

Efforts for Uniformity 

Special effort was made to keep the conditions uniform in all 
the classes. Each teacher was requested to give no drill during 
the experiment in the line of work being tested, since one purpose 
of the work was to measure as accurately as possible the amount 
of improvement her children could make with this special method 
of work in the exact time allotted for the practice. A fine spirit 
of cooperation prevailed which insured compliance with this 
request. No mention was made to the children about practicing 
for fear it might act as a suggestion. In a few cases children 
became so enthusiastic that they did some work at home. How- 
ever, they bad none of the sheets that were used in the experi- 
ment and so had to devise their own means for any practice car- 
ried on. This can only be taken as an indication of the interest 



' This was perhaps an error of administration, but its influence on 
any result to be stated was very slight; hence, having begun in that 
way, I have not thought it worth while to undergo the labor of com- 
puting the results separately for those who were present at every single 
practice-period. 



The Administration of the Experiments ii 

displayed and must be considered, as I have already said, as a 
factor that can not be completely controlled in a school experi- 
ment. 

Perhaps the greatest difficulty in carrying on the tests uni- 
formly for all classes was met in trying to give the instructions 
and appeals to different classes in such a way as to cause them 
to react with an equal amount of enthusiasm for the work. At 
all times every incentive was used that would seem to cause the 
children to surpass their previous records. When one bears 
in mind that the writer was the regular supervisor equally ac- 
quainted with all the classes, controlled by no preconceived 
notions of results, and interested in the experiment as a piece of 
accurate school work, it will be seen that the conditions would 
naturally be kept as nearly uniform in all classes as is possible 
in such a piece of work as has been done. 

Some one may express surprise at not finding here a printed 
set of instructions given orally or read to all classes in the same 
way to insure uniformity. However, the author aimed not so 
much at uniformity in kind and amount of instruction as at uni- 
formity of understanding and enthusiasm on the part of the dif- 
ferent classes for the task in hand. While this plan introduced 
again the judgment of the writer to determine uniformity of un- 
derstanding and enthusiasm, he feels that, even so, less substan- 
tial variability resulted in the total reactions of the classes meas- 
ured, than if the same instructions had been uniformly read or 
given orally at all times. In making this effort for uniformity of 
understanding and enthusiasm, no great variations in instructions 
and appeals were necessary ; and, so far as could be judged, much 
the same intensity of effort obtained in the performance of any 
class from day to day, and in the performance of different 
classes. 



CHAPTER II 
IMPROVEMENT IN THE GROUP AS A WHOLE 

Addition 

This chapter consists of a consideration of the results of the 
practice already described, in so far as they pertain to the group 
as a whole. The large problem to be presented is the amount 
of improvement in the functions tested, made by these school 
children of the third and fourth year in about one hour of drill 
with the practice experiment as the method of class work. 

To measure the improvement in a group of persons subjected 
to a given amount of practice in any phase of activity, it is 
necessary to devise methods and materials which will exercise 
the function in question and to obtain exact and comparable 
measures of their ability at the beginning and end of the prac- 
tice. The methods and material have been discussed in a pre- 
vious chapter. We shall proceed to a presentation of the data 
which give the initial and final ability of the entire group and 
hence the basis for measuring any change which resulted from 
the practice. The clearest presentation of the results of this 
study would demand that all the class records be published in 
detail. But since these records alone would require more than 
half the space ordinarily allotted to such a report as this, only 
a sample record can be published. However, from this record 
the reader can obtain a thorough understanding of the exact 
sources from which all summaries presented later were obtained. 
The table is a complete record of one class in addition in which 
all the practice-periods were 15 minutes in length. Such a record 
is presented because it affords the best opportunity to see the 
progress from one period of practice to the next. 

Table I shows that in the initial practice-period boy A added 
85 columns, 81 of which (or 95 per cent) were correct. In the 
second period he added 105 columns, 97 of which were correct. 
In the third period he added no columns, 102 of which were 

12 



Improvement in the Group as a Whole 



13 



TABLE I 

Record of Class I in Five Fifteen-minute Practice-periods 
IN Addition 



Indi- 


Initial 


2nd 


3rd 


4th 




Final 




Gain 




vid- 


Practice- 


Practice - 


Practice - 


Practice - 


Practice- 








uals 


Period 


Period 


Period 


Period 


Period 










































Per 

cent 




Boys 


S. 


C. 


%c. 


S. 


C. 


S. 


C. 


S. 


C. 


S. 


C. 


%c. 


Gross 


Ac. 


A 


85 


81 


95 


105 


97 


110 


102 


115 


107 


118 


116 


98 


35 


43 


4- 3 


B 


38 


30 


79 


39 


27 


44 


32 


46 


38 


61 


43 


70 


13 


43 


— 9 


C 


83 


81 


98 


80 


53 


85 


61 


98 


74 


92 


77 


83 


—4 


—5 


—15 


D 


40 


27 


68 


39 


31 


48 


36 






57 


34 


42 


7 


26 


—26 


E 


49 


42 


89 


57 


51 


64 


59 


76 


64 


80 


67 


83 


20 


43 


— 6 


F 


80 


74 


93 


90 


89 


101 


99 


119 


117 


112 


112 


100 


38 


51 


+ 7 


G 


51 


49 


96 


68 


67 


75 


71 


89 


86 


98 


97 


99 


48 


98 


-1- 3 


H 


58 


51 


88 


49 


41 


54 


45 


54 


46 


57 


43 


75 


8 


16 


—13 


I 


48 


46 


98 


65 


49 


55 


54 


52 


50 


59 


57 


97 


11 


24 


— 1 


J 


37 


26 


70 


44 


39 


43 


40 


48 


45 


59 


50 


83 


24 


92 


+ 15 


K 


89 


82 


92 


40 


82 


99 


92 


112 


104 


96 


91 


95 


9 


11 


+ 3 


L 


112 


109 


97 


128 


127 


138 


136 


140 


132 


156 


148 


95 


39 


36 


— 2 


M 


42 


38 


90 


47 


42 


51 


45 


49 


42 


51 


47 


92 


9 


24 


+ 2 


N 


30 


27 


90 


41 


40 


45 


44 


47 


44 


50 


45 


90 


18 


67 








24 


15 


63 


35 


19 


52 


17 


74 


25 


51 


27 


53 


12 


80 


—10 


P 


50 


49 


98 


60 


57 


76 


75 


80 


77 


95 


91 


95 


42 


86 


— 3 


Q 


37 


29 


78 


41 


31 


46 


37 


28 


20 


48 


39 


81 


10 


36 


+ 3 


R 


24 


17 


71 






40 


25 


39 


30 


37 


27 


75 


10 


59 


-1- 2 


s 


33 


27 


82 


32 


30 


37 


32 


55 


47 


48 


39 


81 


2 


7 


— 1 


Girls 
































A 


19 


5 


74 


24 


4 


29 


6 


31 


5 


34 


15 


47 


10 


200 


—27 


B 


42 


31 


74 


38 


37 


24 


22 


22 


20 


52 


48 


83 


12 


39 


+ 9 


C 


46 


37 


80 


50 


38 


55 


44 


61 


52 


61 


48 


79 


11 


30 


— 1 


D 


30 


18 


60 


38 


30 


34 


26 


37 


26 


37 


32 


87 


14 


78 


-1-27 


E 


32 


31 


97 


44 


41 


56 


48 


67 


64 


73 


69 


95 


38 


123 


— 2 


F 


19 


16 


84 


43 


41 


50 


45 


55 


47 


78 


77 


99 


61 


381 


-f-15 


G 


26 


20 


75 


44 


30 


33 


19 


49 


32 


36 


25 


69 


5 


25 


— 6 


H 


45 


39 


87 


54 


47 


61 


57 


84 


79 


96 


85 


89 


46 


118 


+ 2 


I 


23 


21 


91 


24 


15 


28 


23 


25 


13 


37 


32 


87 


11 


52 


— 4 


J 


35 


25 


71 


40 


35 


53 


48 


60 


55 


64 


59 


92 


34 


136 


+ 21 


K 


56 


52 


93 


71 


66 


54 


54 


68 


67 


63 


61 


97 


9 


17 


+ 4 


L 


25 


18 


72 


33 


22 


38 


33 


48 


35 


40 


37 


93 


19 


106 


+ 21 


M 


32 


26 


81 


45 


36 


49 


47 


64 


63 


70 


65 


90 


39 


150 


+ 9 


N 


26 


19 


73 


36 


30 


35 


30 


48 


33 


52 


36 


70 


17 


89 


— 3 





75 


74 


99 


80 


80 


83 


77 


89 


79 


96 


89 


93 


15 


20 


— 6 


P 


37 


36 


97 


40 


40 


41 


39 


45 


43 


40 


36 


90 








— 7 


Q 


37 


35 


95 


30 


26 


35 


34 


37 


30 


49 


44 


90 


9 


26 


— 5 


R 


45 


37 


82 


56 


54 


59 


55 


62 


58 


71 


65 


92 


28 


76 


+ 10 


S 


32 


30 


94 


40 


35 


40 


38 


40 


39 


48 


41 


85 


11 


37 


— 9 


T 


34 


31 


91 


38 


35 


48 


42 


56 


47 


56 


41 


75 


10 


32 


—16 


1 


2 


3 


4 


6 


6 


7 


8 


9 


10 


// 


IS 


IS 


14 


IS 


16 



S.=Problems solved. 
C.=Problenis correct. 
%C.=Per cent of problems correct. 
Ac.=Accuracy. 

correct. In the fourth period he added 115 columns, 107 of 
which were correct. In the final period he added 118 columns, 
116 of which (or 98 per cent) were correct. His gross gain was 
35 columns. That is, he added correctly 35 more columns in 
the final fifteen-minute period than in the initial fifteen-minute 



14 Practice in the Case of School Children 

period. This gross gain of 35 columns was on an initial ability 
of 81 columns; and hence means a gain per cent of 43. In the 
initial fifteen-minute period he add 95 per cent of his columns 
correctly ; and in the final fifteen-minute period he added 98 per 
cent of his columns correctly which made a gain of 3 per cent 
in accuracy. Each of the columns of the table has been given 
a number (printed at the bottom) by which it will be referred 
to in the future discussion. 

Referring to these small figures in italics at the bottom of 
the columns, the reader can gain a complete interpretation of 
the entire record from the following explanation, bearing in 
mind also that the letter " S " at the top of a column designates 
the entire number of problems solved, " C " the number of prob- 
lems correct, " %C," the per cent of problems worked correctly, 
"Ac," accuracy, and that the word " Gain " at the head of the 
last three columns is to be taken with each. Column / gives 
the individuals, boys and girls separate. Column 2 gives the 
entire number of problems solved in the initial 15-minute period; 
column 5 gives the number of problems that were correct in the 
same period ; and column 4 gives the per cent of problems 
worked correctly, or the per cent of accuracy in the initial period. 
The percentages of accuracy in column 4 were found by dividing 
the number in column 5 by the corresponding number in column 
2. Columns 5 and 6, 7 and 8, 9 and 10 give the entire number 
of problems worked and the number right in the three inter- 
vening 15-minute periods. These pairs of columns correspond 
to 2 and 5 in the initial practice-period. The per cent of accuracy 
for these intervening tests was not necessary for later discussion 
and so was omitted. Columns //, 12 and Jj of the final period 
correspond to columns 2, 5, and 4 in the initial period. Column 
14 gives the gross gain in number of problems correct or the 
absolute gain, and is the algebraic difference between the num- 
ber in column j and the corresponding number in column 12. 
Column 75 gives the gain expressed in hundreths obtained from 
dividing the number in column 14 by the corresponding number 
in column j. Column 16 is the gross gain in accuracy expressed 
in per cent of problems correct and is the algebraic difference 
between the number in column 4 and the corresponding number 
in column jj. It is to be noted that this last column is not a 
gain per cent as is column 75, but the gross gain in accuracy 



Improvement in the Group as a Whole 15 

expressed in a per cent. A minus sign indicates a loss wherever 
used. 

Every other class record in addition used in this study corre- 
sponds closely to this one in the categories of initial record, 
final record, per cent of accuracy, gross gain, percentile gain, and 
per cent gained in accuracy. The only difference in other class 
records results from the fact that the intervening periods of 
practice were of different lengths, some being 22^/4 minutes, 
some 6 minutes, and some 2 minutes in length. 

For the entire addition experiment, there are 21 class records. 
These complete records are placed on file in Teachers College 
where they are accessible to any one wishing to use them. 

Initial Ability 

The first group of facts to be considered in arriving at a solu- 
tion to the problem with which this chapter deals, provides data 
which give a measure of the ability of the entire group in addi- 
tion at the beginning of practice. All the individual records of 
" problems added correctly " in the initial period of which those 
in column j of Table I are samples, have been distributed, class 
by class, boys and girls separate. A summary of these facts 
affords the data necessary to show the distribution, central ten- 
dency, and variability of the group as a whole, as well as the 
reliability of the measures used. 

Table II gives a distribution of the number of problems 
worked correctly in the initial fifteen-minute period by 732 
fourth-year children. The number of problems worked is given 
in groups of 5. The table shows that 3 children (or 0.4 per cent 
of the entire group) worked from o to 4 problems correctly in 
fifteen minutes; 50 children (or 6.8 per cent) worked from 5 to 9 
problems correctly; 80 children (or 10.9 per cent) did from 10 to 
14 problems, etc. The figures show that the distribution is de- 
cidedly skewed toward the high end. The table shows the range 
in number of problems added correctly by these fourth-year 
children to be from ' o to 4 ' problems up to ' 105 to 109 ' prob- 
lems. The median number of problems worked correctly is 23.3 ; 
that is, there are as many children who worked 23.3 problems 
or more as there are who worked 23.3 problems or less. The 
variability of the group is further shown by the P. E., 7.8, a 



i6 



Practice in the Case of School Children 



TABLE II 

NtTMBER OF Problems Added Correctly in the Initial Fifteen- 
minute Period 



Problems added 
correctly .... 




to 

4 


5 

to 
9 


10 
to 
14 


15 

to 
19 


20 
to 
24 


25 

to 
29 


30 
to 
34 


35 

to 
39 


40 
to 
44 


45 

to 
49 


50 
to 
54 


55 

to 
59 


Number of indi- 
viduals 


3 


50 


80 


133 


130 


111 


81 


57 


33 


14 


13 


7 


Per cent 


.4 


6.8 


10.9 


18.1 


18 


15.2 


11 


7.8 


4.5 


1.9 


1.8 


.9 


TABLE 11— Continued 


Problems added 
correctly .... 


60 

to 
64 


65 

to 
69 


70 
to 
74 


75 

to 
79 


80 
to 

84 


85 
to 

89 


90 

to 
94 


95 

to 
99 


100 

to 

104 


105 

to 

109 


Total 


Number of indi- 
viduals 


4 


3 


7 





3 


1 


1 








1 


732 


Per cent 


.5 


.4 


.9 





.4 


.1 


.1 








.1 


100 



Median 
25 Percentile 
75 Percentile 
P.E. 

P-E- t.-obt. Av. 



23.3 

16.5 

32.1 

7.8 

.29 



problems 



number which when subtracted from and added to the median, 
gives the approximate Hmits within which the middle fifty per 
cent of the class falls. In other words, when the lowest twenty- 
five per cent and the highest twenty-five per cent of the class 
are cut out, the limits within which the remaining middle fifty 
per cent falls, are 16.5 columns and 32.1 columns. The per cents 
in the table show that 81 per cent of the children ranged from 
10 to 39 problems; 51 per cent from 15 to 29 problems. 

Accuracy 

In order to define more exactly the initial ability of this group 
of children in addition it is necessary to know the degree of 
accuracy of their performance. The accuracy for each child 
was determined by dividing the number of problems worked cor- 
rectly by the entire number worked. Just what these per cents 
of accuracy are the reader may see from Table I, column 4. 



Improvement in the Group as a Whole 



17 



The following table is a distribution of the per cents of accuracy 
of the entire group of 732 children, of which those per cents 
in column 4 of Table I form a part. 



TABLE III 

Per Cent of Problems Added Correctly in the Initial 
Fifteen-minute Period 



Per cent of accuracy. . . 


6 
to 
10 


11 

to 
15 


16 
to 
20 


21 
to 
25 


26 
to 
30 


31 
to 
35 


36 
to 
40 


41 
to 
45 


46 
to 
50 


51 
to 
55 


Individuals 


1 


1 





2 


3 


9 


13 


16 


22 


17 






Per cent of group 


.1 


.1 





.3 


.4 


1.2 


1.8 


2.2 


3 


2.3 



TABLE III— Continued 



Per cent of accuracy. . . 


56 
to 
60 


61 
to 
65 


66 
to 
70 


71 
to 
75 


76 
to 
80 


81 

to 

85 


86 
to 
90 


91 
to 
95 


96 
to 
100 


Total 


Individuals 


25 


46 


58 


84 


100 


81 


108 


76 


70 


732 


Per cent of group 


3.4 


6.3 


8 


11.5 


13.7 


11.1 


14.8 


10.4 


9.6 


100% 



Median 


79 . per cent 


25 Percentile 


68. 


75 Percentile 


89. 


P.E. 


10. 


P-^- t.-obt. Av. 


.38 



Table III gives the per cents of accuracy in groups of five. 
It shows that 70 children (or 9.6 per cent of the entire group) 
added correctly from 96 to 100 per cent of the entire number of 
problems which they worked ; 76 added correctly from 91 to 
95 per cent of the problems which they worked, etc. These 
figures show that the distribution is skewed toward the low end. 
The range in per cent of problems added correctly is from 6 to 
100. The median per cent of accuracy is 79, which shows that 
there were as many children who added correctly four-fifths or 
more of their problems as there were who added correctly four- 
fifths or less. The upper 25 per cent of the class added with an 
accuracy of 89 per cent or more ; the upper 60 per cent of the 
class added with an accuracy of 68 per cent or less. The range 



iB Practice in the Case of School Children 

of accuracy for the middle 50 per cent of the class was from 
68 per cent to 89 per cent. 

By combining now the data derived from Tables II and III, 
the initial ability of the group can be defined both in terms of 
amount and accuracy of performance. We can give the clearest 
idea of the initial performance of this group in addition by say- 
ing that the median ability in number of problems worked cor- 
rectly in fifteen minutes was 23.3 problems, which was 79 per 
cent of the entire number of problems worked. Or, taking as a 
basis the entire number of problems worked instead of those 
worked correctly, one can say that the chances are that one child 
in any two picked at random would be able to add 29.5 columns 
or more in 15 minutes with an accuracy of 79 per cent or more. 

Gross Gain in Number of Problems Correctly Added 

We are now ready to consider the gain made in problems cor- 
rectly added by this group of fourth-year children in the course 
of seventy-five minutes of drill with the practice experiment as 
the method of class work.* This gain will be considered in two 
ways, absolute gain and percentile gain. The present considera- 
tion has to do with the absolute gain. It will be recalled that 
the initial test and the final test for all classes were of equal 
length, 15 minutes, and that between the initial and final tests 
all classes practiced for the same length of time, 45 minutes. 
It is now our problem to measure the amount of improvement 
that the entire group showed in the final 15 minutes of practice 
over the initial 15 minutes. 

Referring again to Table I, columns 5, 12 and 14, the reader 
will understand clearly the exact measures that are involved in 
this part of the discussion. It will be seen that boy A in the 
initial test had 81 problems correct; in the final test he had 116 
problems correct which gave him an absolute gain of 35 prob- 
lems in the final 15 minutes over the number of problems worked 
correctly in the initial 15 minutes. So, boy B gained 13 prob- 
lems, girl A 10 problems, etc. All of these gross gains were 



* The gain is measured for only 60 minutes of practice. The initial 
period is 15 minutes and the final period is 15 minutes. The record 
for each of these periods gives the adding rate at the middle of each 
period. Hence the time of practice that is measured is from the middle 
of the initial period to the middle of the final period, or (to minutes. 



Improvement in the Group as a Whole 



19 



distributed in a table, class by class, boys and girls separate. 
Only a summary of this distribution for the entire group is pre- 
sented here. 

TABLE IV 

Gross Gain, in Number op Columns Added Correctly in Fifteen 
Minutes, Resulting from Sixty Minutes of Practice 



Columns gained. . . 


—15 

to 
—12 


—11 

to 

—8 


—7 

to 

—4 


—3 
to 



1 

to 
4 


5 

to 
8 


9 
to 
12 


13 
to 
16 


17 
to 
20 


21 
to 
24 


25 
to 
28 


Individuals 


5 


11 


28 


63 


93 


105 


111 


90 


75 


29 


29 


Per cents 


.7 


1.5 


3.8 


8.6 


12.7 


14.3 


15.2 


12.3 


10.2 


4 


4 



TABLE lY— Continued 





29 


33 


37 


41 


45 


49 


53 


57 


61 


65 




Columns gained. . . 


to 


to 


to 


to 


to 


to 


to 


to 


to 


to 


Total 




32 


36 


40 


44 


48 


52 


56 


60 


64 


68 




Individuals 


25 


24 


16 


5 


7 


5 


2 


3 


4 


2 


732 


Per cents 


3.4 


3.3 


2.2 


.7 


.9 


.7 


.3 


.4 


.5 


.3 





Median 


10.7 columns 


25 Percentile 


3.6 


75 Percentile 


18.8 


P.E. 


7.6 


^•^- t.-obt. Av. 


.28 



Table IV gives a summary of the distribution of the gross 
gain in number of columns added correctly in fifteen minutes 
resulting from 60 minutes of practice. The number of problems 
gained is given in groups of 4. The table reads : " Five children 
(or 0.7 per cent of the entire group) gained from — 15 to — 12 
problems." This means, of course, that they lost. The other 
end of the table reads: "Two children (or 0.3 per cent of the 
entire group) gained from 65 to 68 problems ; four children (or 
0.5 per cent of the group) gained from 61 to 64 problems, etc." 
The figures show that the distribution is skewed toward the high 
end. The range for the group is from 15 to 12 problems lost 
to 65 to 68 problems gained. The median gain is 10.7 columns, 
which means that the median increase in ability was such that 
10.7 more columns were added correctly in 15 minutes at the 
end of practice than in 15 minutes at the beginning of practice. 



20 Practice in the Case of School Children 

It also means there were just as many children who gained 10.7 
columns or more, as there were who gained 10.7 columns or less. 
The members of the upper 25 per cent of the group gained 
18.8 columns or more while the members of the lower 25 per 
cent of the group gained 3.6 columns or less. This leaves the 
range for the middle fifty per cent of the class from 3.6 columns 
to 18.8 columns gained. The percentages in the table show that 
65 per cent of the entire group gained from i to 20 problems 
and that a larger per cent gained from 9 to 12 columns, than 
any other group of four columns. 

Relative Gain in Addition From Sixty Minutes of Practice 

Gain may be expressed in absolute terms, as in the last section ; 
or it may be expressed in the per cent which the gross gain is 
of the original quantity on which it was gained. We are so 
accustomed to think of gain or improvement in relative terms 
that such terms often become a more satisfying measure of the 
change in question than the absolute terms. We found in the 
last section that the median gain in number of problems added 
correctly in 15 minutes was 10.7 columns. It is now the problem 
to find what per cent of the initial ability this gross improve- 
ment was. 

Referring again to Table I, column i^, the reader will clearly 
understand the source from which the present data are obtained. 
Boy A made a gross gain of 35 problems ; that is, he worked 
correctly 35 more problems in the last fifteen minutes than in 
the initial fifteen minutes. Or he made a gain of 35 problems 
on an initial ability of 81 problems, or 43 per cent. So boy B 
gained 43 per cent, girl A gained 200 per cent and girl B, 39 
per cent, etc. In almost every case these per cents are of a dif- 
ferent initial ability, and to have their proper significance they 
need to be considered in relation both to the initial ability and 
the gross gain of the individual whose performance they repre- 
sent. That is, the same gain per cent for two individuals does 
not necessarily indicate that their gain was equally meritorious. 
Taking the cases of boys A, B and E, who have gained the same 
relative amount, we would be justified in saying that A's 43 per 
cent gain represents a much higher efficiency of performance 
that that of B or E. Likewise we feel certain that L's 36 per 



Improvement in the Group as a Whole 2 1 

cent of gain is far more noteworthy than Q's 36 per cent, and 
it is plainly evident that G's 98 per cent does not mean almost 
twice as meritorious a gain as that of F. We are badly in need 
of a more equitable means of measuring relative gain than is 
afforded by per cents, but until such a means is perfected we 
shall have to continue using the one now in vogue. Such consid- 
erations, however, have to do especially with individual differ- 
ences which will be considered in a later portion of this study. 
All the per cents which correspond to those of column 15, 
Table I, were distributed class by class, boys and girls separate, 
but only a summary of this table is presented here, from which 
an adequate notion of the central tendency and variability of 
the group in relative gain can be explained. 



TABLE V 



Gain Per Cent in Columns Added Correctly in Fifteen Minutes 
Resulting from Sixty Minutes of Practice 


Gain, per cent . . . 


-89 
to 
-75 


-74 
to 
-60 


-59 
to 
-45 


-44 
to 
-30 


-29 
to 
-15 


-14 
to 



1 

to 
15 


16 

to 
30 


31 
to 
45 


46 
to 
60 


61 

to 

75 


76 
to 
90 

54 


91 
to 
105 

40 


106 
to 
120 


Individuals 


3 


1 


6 


11 


20 


61 


66 


97 


85 


92 


80 


20 






Per cents 


.4 


.1 


.8 


1.5 


2.7 


8 3 


.9 


13.2 


11.6 


12.6 


10.9 


7.4 


5.5 


4 







TABLE V— Continued 



Gain, per cent.. . 



121 
to 
135 



136 
to 
150 



151 
to 
165 



166 
to 
180 



181 
to 
195 



196 
to 
210 



211 
to 
225 



226 241 
to to 
240 255 



256 
to 
270 



271 
to 
285 



286 
to 
300 



301 

to 



Total 



Individuals 



14 



21 



9 



1 







10 



732 



Per cents. 



1.9 



2.9 



1.2 



.9 



.3 



1.1 .1 



1.4 



100 



Median 48 per cent 

25 Percentile 18 

75 Percentile 83 

P.E. 32.5 



In Table V the relative gains are distributed in groups of 
15. The range is from a loss of 89 per cent to a gain of 300 
per cent or more. The mode is in the group 61 to 75 per cent. 
The median gain per cent is 48, which means that there were as 



3 2 Practice in the Case of School Children 

many children who gained more than 48 per cent as there were 
who gained less than 48 per cent. The upper 25 per cent of the 
group gained 83 per cent or more while the lower 25 per cent 
gained 18 per cent or less, which leaves the range for the 
middle 50 per cent of the group from a gain per cent of 18 to 
83. The percentages in the table show that 14 per cent of the 
entire group lost as a result of the practice, and that 56 per cent 
of the entire group gained from 16 to 90 per cent. Again bear- 
ing in mind the wide variability of the class, we can say that 
the central tendency of the relative gain of the group is a gain 
of 48 per cent. 

Gross Cain in Accuracy in Addition 

To determine the complete effect of the practice upon this 
group one other factor remains to be considered, viz., the influ- 
ence of the practice upon the accuracy of making the associations 
involved. It will be recalled from the instructions given to 
the classes just before each i)ractice-period that two factors were 
emphasized at all times during the practice as constituent ele- 
ments of the efficiency of the performance, speed and accuracy. 
We have seen that the group as a whole worked with enough 
increase of speed to add correctly 10.7 more columns in the final 
fifteen-minute period than in the initial period, which meant an 
increase in speed of about fifty per cent. The question now 
arises, was this increase in speed accompanied by an increase 
in accuracy, or was it at the expense of accuracy, or did the 
group as a whole add with as good a chance of getting a correct 
answer at the end of the practice as at the beginning? 

Vox an understanding of the data on which the answer to this 
question is based, the reader is referred again to Table 1. columns 
4, 75, and 16. The degree of accuracy of boy A at the beginning 
of practice was 95 per cent ; at the end of practice his percentage 
of accuracy was 98 per cent. Hence his gross gain in accuracy 
expressed in percentage of answers that were correct was 3. It 
is to be noted that this is a gross gain and not a gain per cent. 
So boy B lost 9 per cent, girl A lost 2y per cent, and girl ?> 
gained 9 per cent. All these per cents were distributed class by 
class, boys and girls separate. A summary of this table will 
suffice to give sufficient data to determine the change in the group 
as a whole, in control tendency and variability. 



Improvement in the Group as a Whole 



23 



TABLE VI 

Gross Gain in Accuracy in Addition Expressed in Per Cent 

OF Answers that Were Correct Resulting from 

THE Sixty Minutes of Practice 



Per cent gained. 


-55 
to 
-51 


-50 
to 
-46 


-45 
to 
-41 


-40 
to 
-36 


-35 
to 
-31 


-30 
to 
-26 

23 


-25 
to 
-21 


-20 
to 
-16 


-15 
to 
-11 


-10 
to 
-6 


-5 

to 
-1 



to 
4 


5 
to 
9 


Individuals 


8 


2 


2 


6 


19 


17 


54 


63 


68 


101 


111 


79 


Per cents 


1.1 


.3 


.3 


.8 


2.6 


3.1 


2.3 


7.4 


8.6 


9.3 


13.8 


15.2 


10.8 









TABLE V 


— Continued 












Per cent gained. 


10 
to 
14 


15 

to 
19 


20 

to 

24 


25 

to 
29 


30 
to 
34 


35 
to 
39 


40 
to 
44 


45 
to 
49 


50 
to 
54 


55 
to 
59 


60 
to 
64 


65 
to 
69 


Total 


Individuals 


61 


47 


28 


17 


6 


8 


2 


5 


1 


3 





1 


732 


Per cents 


8.3 


6.4 


3.8 


2.3 


.8 


1.1 


.3 


.7 


.1 


.4 





.1 





Median 


-.4 


25 Percentile 


-12.4 


75 Percentile 


9.2 


P.E. 


10.8 



per cent 



In Table VI, the gross gains in accuracy expressed in per 
cents are given in groups of 5. Negative signs indicate loss. 
The range in the change in accuracy effected by the practice is 
from a loss of 55 per cent to a gain of 69 per cent. The table 
reads, " Eight children (or i.i per cent) of the entire group lost 
from 55 to 51 per cent, etc." The figures indicate a curve which 
approaches close to the normal curve of distribution. The mode 
is at o to 4 per cent gain. The median is .4 per cent loss, which 
shows that there were as many children who lost .4 per cent or 
more in accuracy as there were who lost .4 per cent or less or 
gained in accuracy. The figures following '' Individuals " show 
that 363 of the children lost in accuracy while 369 children either 
maintained their initial proficiency for accuracy or worked with 
greater chance of getting a correct answer. The upper 25 per 
cent of the group gained 9 per cent or more in accuracy while 
the lower 25 per cent lost 12 per cent or more, which gives the 
range for the middle 50 per cent of the group from a loss of 
12 per cent to a gain of 9 per cent. 



24 Practice in the Case of School Children 

Summary 

The initial performance in addition of the group of 732 fourth- 
year children showed a median ability on their part to add cor- 
rectly in fifteen minutes 23.3 columns such as were used in the 
practice. Their median accuracy in this initial performance 
was found to be about 80 per cent, which shows that the median 
number of problems added regardless of the answer obtained 
was about 30 columns. From these two central tendencies we 
can say that the median initial ability of this group of 732 fourth- 
year children in addition was such that they could add correctly 
23.3 columns, such as were used in the test, with an accuracy 
of 80 per cent. 

The change effected by the practice of 60 minutes m the group 
as a whole was an improvement in speed such that they could 
add 10.7 more columns correctly in the final 15 minutes of the 
practice than in the initial 15 minutes, while the change in accur- 
acy was so small as not to need consideration. 

Division 

The experiment in division, with children in the second half 
of the third year and the first half of the fourth year, was con- 
ducted on the same general plan as the one in addition with 
fourth-year children, except that the initial and final periods of 
practice were 10 minutes each instead of 15, and that the entire 
time spent in the practice was 60 minutes instead of 75. The 
intervening 40 minutes between the initial and final practice- 
periods were divided for different groups of classes into periods 
of 20 minutes, 10 minutes and 2 minutes. This plan of time re- 
quired 4 school days to complete the experiment when the inter- 
vening time was divided into 20-minute periods. 6 consecutive 
school days wiicn the intervening time was divided into lo-minute 
periods, and 22 consecutive school days when the intervening 
40 minutes were divided into periods of 2 minutes each. The 
method of conducting the practice was described in Chapter I, 
and a sample sheet of the material used may be found on page 
<)8. It is necessary to remember that in scoring these division 
combinations a credit of one was given for each combination 
if the quotient and remainder were both correct. If either was 
wrong, no credit was given. In other words, the score was 



Improvement in the Group as a Whole 25 

found by deducting one from the entire number of combina- 
tions worked for each one that was incorrectly worked. 

In Table VII, an exact class record is given of a class for 
which the intervening 40 minutes were divided into periods of 
ten minutes each. This arrangement made all the practice- 
periods of this class of equal length. Such a record is presented 
because it gives samples of all data to be used in this discussion 
and affords the best opportunity to note the change in ability 
from day to day. In all, eighteen classes took part in the division 
experiment, which means that there were eighteen such records 
as the one here presented, from which the data to be presented 
in the following discussion were obtained. Six of these 18 
records involve exactly the same number of practice-periods as 
the one given in Table VII. Six others have but two practice- 
periods between the initial and the final periods and so arc more 
brief. The other 6 have 20 practice-periods between the initial 
and final periods and hence occupy more than three times the 
space occupied by the one given in Table VII. Six hundred 
six third- and fourth-year children took part in the experiment, 
in the course of which about 6500 papers were scored and en- 
tered in these 18 records. These can not be printed here, but 
they are placed on record in Teachers College where any one 
may use them. The following table gives sufficient data to enable 
the reader to understand the exact sources from which the 
summaries that are to be presented later were ol)tained. 

Table VII reads as follows, " Boy B in the initial practice- 
period worked 70 combinations, 63 of which were correct, or 
90 per cent of them. In the second practice-period he worked 
72 combinations with 72 correct. In the third practice he 
worked 80 combinations with 80 correct. In the fourth prac- 
tice he worked 87 combinations with 83 correct. In the fifth 
practice-period he worked 99 combinations with 92 correct. In 
the final practice-period he worked 119 combinations with 112 
correct, or 94 per cent of them. His gain in number of com- 
binations worked correctly was 49, his gain per cent 78, and 
his gain in accuracy expressed in per cent was 4 j^er cent." So 
for any other individual. In the following discussion the 
quantities mentioned above will be referred to by the numbers 
in italics at the bottom of the columns. The mctliod of finding 
the per cent of accuracy, the gain per cent in number of problems 



26 



Practice in the Case of School Children 



TABLE VII 

Record of Class VII in Six Ten-minute Practice-periods in 
Division 



Indi- 


Initia 




2nd 


3 


-d 


4th 


5 


th 


Final 






Gain 




vid- 


Practice- 


Practice- 


Practice - 


Practice - 


Practice - 


Practice- 






uals 


Tt^^inA 


-parir^A 


•Po^-.r^A 


Po^;^,4 


p«,.i«^i 


p««;,^^ 






IT 






r^er 




xer 




-L er 




x^er 


" 


i: 


Cll-Jl 






Pai- 
































x^er 




Boys 


S. 


C. %C. 


s. 


C. 


s. 


C. 


s. 


C. 


S. 


C. 


s. 


C. %C. 


Gross 


cent 


Ac. 


A 


55 


54 


98 


60 


58 


84 


82 


77 


75 


76 


75 


74 


71 


96 


17 


30 


— 2 


B 


70 


63 


90 


72 


72 


80 


80 


87 


83 


99 


92 


119 


112 


94 


49 


78 


+ 4 


C 


27 


25 


93 


45 


38 


55 


51 


59 


56 


66 


60 


70 


66 


94 


41 


168 


+ 1 


D 


16 


12 


75 


14 


14 


25 


24 


25 


24 






18 


17 


95 


5 


63 


+ 20 


E 


77 


77 


100 


99 


98 


106 


106 


112 


112 


133 


133 


153 


151 


99 


74 


96 


— 1 


F 


27 


21 


78 


28 


18 


19 


13 


30 


23 


36 


32 


45 


42 


93 


21 


100 


+ 15 


G 


31 


25 


81 


36 


31 


46 


43 


41 


35 


38 


31 


49 


42 


86 


17 


68 


+ 5 


H 


50 


49 


98 


51 


48 


61 


61 


56 


53 


55 


53 


65 


63 


97 


14 


29 


— 1 


I 


62 


62 


100 


62 


62 


69 


69 


69 


67 


60 


60 


68 


68 


100 


6 


10 





J 


34 


30 


88 


48 


44 


58 


57 


54 


52 


52 


50 


53 


52 


98 


22 


73 


+ 10 


K 


18 


17 


95 


36 


33 


54 


50 


49 


44 


77 


70 


63 


59 


94 


42 


40 


— 1 


L 


30 


19 


63 


35 


25 


46 


33 


64 


58 


40 


36 


37 


35 


95 


16 


84 


+ 32 


M 


90 


86 


96 










104 


98 


114 


113 


128 


127 


99 


41 


48 


+ 3 


N 


69 


68 


99 


62 


59 


83 


81 


78 


75 


77 


77 


92 


89 


98 


21 


31 


— 1 





43 


38 


89 


53 


45 


69 


65 


61 


58 


56 


53 


63 


60 


95 


22 


58 


+ 6 


P 


27 


27 


100 


27 


27 


33 


32 


31 


27 


28 


28 


30 


27 


90 








—10 


Q 


78 


74 


95 


89 


86 


109 


99 


104 


104 


108 


98 


107 


98 


91 


24 


32 


— 4 


K 


37 


37 


100 


58 


58 


67 


67 


62 


60 


69 


67 


77 


77 


100 


40 


105 





S 


48 


44 


92 






63 


57 


55 


52 


66 


66 


77 


77 


100 


33 


75 


+ 8 


T 


67 


66 


99 


88 


86 


103 


101 


102 


101 


99 


95 


112 


111 


99 


45 


68 





U 


54 


51 


94 


57 


54 


92 


68 


69 


67 


62 


62 


64 


64 


100 


13 


25 


+ 6 


V 


35 


32 


92 






40 


37 


35 


32 


44 


41 


56 


54 


97 


22 


69 


+ 5 


w 


27 


25 


92 


27 


25 


43 


39 


38 


35 


36 


36 


39 


39 


100 


14 


56 


+ 8 


Girls 




































A 


14 


11 


79 


29 


21 


37 


29 


31 


19 






38 


31 


82 


20 


55 


+ 3 


B 


24 


23 


96 


37 


36 


30 


29 


35 


32 


45 


45 


50 


50 


100 


27 


117 


+ 4 


C 


39 


36 


93 


39 


38 


52 


51 


53 


49 


50 


48 


57 


53 


93 


17 


47 





D 


50 


46 


92 


59 


56 


69 


68 


62 


62 


51 


51 


81 


80 


99 


34 


74 


+ 7 


E 


56 


54 


97 


61 


58 


61 


61 


65 


65 


55 


53 


66 


62 


94 


8 


15 


— 3 


F 


52 


49 


94 


54 


52 


71 


71 


66 


63 


50 


48 


76 


73 


96 


24 


49 


+ 2 


G 


46 


41 


89 


49 


46 


51 


49 


50 


47 


49 


46 


66 


61 


91 


10 


24 


+ 2 


H 


42 


39 


93 


53 


51 






68 


68 


65 


64 


79 


77 


98 


38 


97 


+ 5 


I 


76 


74 


98 


20 


19 


83 


83 


78 


77 


81 


79 


110 


107 


97 


33 


45 


— 1 


J 


47 


46 


98 


50 


48 


69 


68 


55 


55 


55 


55 


66 


61 


97 


18 


37 


— 1 


K 


10 


8 


81 


7 


2 


25 


24 


14 


7 


28 


26 


20 


13 


65 


5 


63 


—23 


L 


72 


69 


96 


71 


69 


93 


92 


93 


93 


90 


87 


98 


96 


98 


27 


39 


+ 2 


M 


45 


45 


100 


55 


54 


66 


62 


77 


76 


63 


54 


71 


67 


95 


22 


19 


— 5 


N 


6f) 


57 


95 


59 


59 


72 


72 


57 


56 


70 


70 


84 


83 


99 


26 


46 


+ 4 


O 


24 


16 


67 


37 


22 


36 


27 


45 


36 


41 


32 


52 


45 


87 


29 


181 


+ 20 


P 


45 


45 


100 


39 


38 


59 


59 


64 


58 


62 


61 


71 


70 


99 


25 


56 


— 1 


Q 


24 


20 


83 


16 


15 


40 


39 


32 


30 


34 


31 


29 


26 


90 


6 


30 


+ 7 


R 


42 


40 


95 


43 


42 


50 


50 


50 


49 


45 


44 


52 


49 


94 


9 


23 


— 1 


S 


23 


20 


87 






29 


29 


36 


25 


33 


32 


37 


36 


98 


16 


80 


+ 1 


T 


26 


23 


89 


44 


41 


56 


53 


51 


50 


43 


42 


56 


56 


100 


33 


143 


+ 11 


U 


30 


16 


46 


30 


16 


37 


29 


32 


23 


37 


32 


38 


36 


95 


30 


125 


+ 49 


/ 


2 


S 


4 


5 


6 


7 


8 


9 


10 


11 


12 


IS 


H 


IB 


16 


17 


18 



S.=Problenis solved. 
C.=Problems correct. 
%C.=Per cent of problems correct. 
Ac.=Accuracy. 



Improvement in the Group as a Whole 



27 



worked correctly, and the gain in accuracy expressed in per 
cent is the same as that used in addition, Table I, to which the 
reader may refer for the method if the meaning of these figures 
is not clear. " S " indicates problems solved, " C " problems 
solved correctly, and "Ac." accuracy. 

Initial Ability 
To determine the initial ability of this group of 606 children 
of last half of third and first half of fourth year in giving the 
results for the division combinations, two factors must be con- 
sidered, — first the number of such combinations they worked cor- 
rectly, and second the accuracy of their performance. The data 
to determine the first factor are the number of combinations 
worked correctly in the initial ten-minute period, a sample of 
which is given in Table VII, column 5. The numbers of com- 
binations answered correctly by the 606 children were distributed 
class by class, boys and girls separate, but only the following 
summary of this distribution table can be presented here. (Table 
VIII.) 

TABLE VIII 



Number of 


Division Combinations Answered Correctly 
Initial Ten-minute Period 


IN 


THE 




Number of 

combinations 



to 
4 


5 
to 
9 


10 
to 
14 


15 
to 
19 


20 
to 
24 


25 
to 
29 


30 
to 
34 


35 
to 
39 


40 
to 
44 


45 

to 
49 

44 


50 
to 
54 

38 


55 

to 
59 

42 


60 

to 
64 

28 

4.6 


65 
to 
69 


Individuals 





23 


33 


65 


56 


52 


51 


57 


45 


15 




Per cent 




3.8 


5.4 


10.7 


9.2 


8.6 


8.4 


9.4 


7.4 


7.3 


6.3 


6.9 


2.5 









TABLE VIII- 


—Continued 












Number of 

combinations .... 


70 
to 
74 


75 

to 
79 


80 
to 

84 


85 
to 
89 


90 
to 
94 


95 
to 
99 


100 
to 
104 


105 
to 
109 


110 
to 
114 


115 
to 
119 


120 
to 
124 


125 
to 
129 


Total 


Individuals 


13 


16 


3 


9 


5 


5 





3 


1 





1 


1 


606 


Per cent 


2.1 


2.6 


.5 


1.5 


.8 


.8 





.5 


.2 





.2 


.2 


100 



Median 


34 . 5 combinations 


25 Percentile 


21.7 


75 Percentile 


52.5 


P.E. 


15.4 


P-E- t.-obt. Av. 


.63 



28 Practice in the Case of School Children 

Table VI 11 shows a distribution (in groups of 5) of the num- 
ber of combinations done correctly in the initial 10 minutes. It 
shows that 23 children (or 3.8 per cent of the entire group) 
did from 5 to 9 combinations, 33 children (or 5.4 per cent of the 
entire group) did from 10 to 14 com])inations, etc. The range 
in number of ccjmjjinations done correctly is from 5 to 126. The 
median number is 34.5 combinations. This means that there 
were just as many children who worked 34.5 combinations or 
more correctly in 10 minutes as there were who worked 34.5 
coml)inations or less. The upper 25 per cent of the group did 
53 combinations or more while the lower 25 per cent did 22 com- 
binations or less. The percentages in the table show that 74 
per cent of the group did from 15 to 59 combinations. The 
variability of the group is also shown by the P. E., 15.4. The 
(lata presented here give 34.5 combinations as the most likely 
true median. 

Acctiracy in Dnnsion 

With what degree of accuracy did this group work the com- 
binations which they attempted in the initial to minutes, is the 
next (|uestion for consideration. The source of the data used 
in answering this question can best be seen in Table VII, column 
./. The per cents of accuracy of which those in column ./ are 
I)art were distributed class by class, boys and girls separate. The 
following table (Table IX) shows a summary from this larger 
tabic. 

Table IX reads as follows from the right side: " Two hundred 
and thirty-seven children (or 39.1 per cent of the entire group) 
worked 96 to lOO per cent of their combinations correctly, etc." 
The ]>er cents are given in groups of 5. The range is from 26 
per cent of accuracy to 100 per cent. The figures show almost 
a right-angle distribution. However, had the per cents been 
scaled more finely at the upper end, the distribution would then 
have shown a very decided skewness toward the lower end. The 
median i)cr cent of accuracy is 93, that is, just as many children 
worked with an accuracy of 93 to 100 per cent as with an 
accuracy of 26 to 93 per cent. The upper 25 per cent of the 
children worked with :m accuracy of 97 per cent or more. The 
lower 25 per cent worked with an accuracy of 85 per cent or less. 



Improvement in the Group as a Whole 29 

TABLE IX 
The Per Cent of Coukect Answeks to the Division Combina- 
tions IN THE Initial Ten-minute Pehiods 



Per cent of correct answers 


26 
to 
30 


31 
to 
35 


36 
to 
40 


41 
to 
45 


46 
to 
50 


51 
to 
55 


56 
to 
60 


61 
to 
65 


Individuals 


1 





1 


3 


7 


5 


7 


12 


Per cent 


.2 





.2 


.5 


1.2 


.8 


1.2 


2 







TABLE IX- 


—Continued 












Per cent of correct answers 


66 
to 
70 


71 
to 
75 


76 
to 

80 


81 
to 

85 


86 
to 
90 


91 
to 
95 


96 
to 
100 


Total 


Individuals 


11 


22 


31 


53 


70 


146 


237 


606 






Per cent 


1.8 


3.6 


5.1 


8.7 


11.6 


24.1 


39.1 


100 







Median 93 per cent 

25 l^ercentile 85 

75 Percentile 97 
P.E. 6 

The upper 63 per cent of these children worked with an accuracy 
of 91 per cent or more. 

With the accuracy of the group determined, we can now 
define the initial ability of the group more exactly by saying 
that it was such that the median number of columns worked 
correctly by the group as a whole was 34.5 combinations, and 
according to the median per cent of accuracy 34.5 combinations 
were 93 per cent of the median number of combinations at- 
tempted. The chances are that one child out of two in the group 
taken at random would do 34.5 combinations correctly in 10 
minutes with an accuracy of 93 per cent. 

Gross Gain in Number of Combinations Worked Correctly 
What gain in ability did this group of 606 third- and fourth- 
year children make in the course of 60 minutes of practice, is 
the next question to be answered.^ This gain has been measured 

" While there were 60 minutes of practice the gain was measured for 
only so minutes. The initial practice-period and the final practice-period 
were 10 minutes each. But the record for each of these periods gives 
the adding rate at the middle of each period. Hence the amount of 
practice whose eflfect is measured is from the middle of the initial 
period to the middle of the final period, or 50 minutes. 



3° 



Practice in the Case of School Children 



both absolutely and relatively. (Xir ])rc.sent consideration is the 
gnjss fjain, wliicii was found for each individual by subtracting 
the number (^f combinations worked correctly in the first lo min- 
utes of practice from the number worked correctly in the final 
lo minutes of practice. 

Jveferring to Table VII, column 16, the reader will sec that 
boy A J^^•lined 17 combinations, boy 15, 49 combinations, etc. The 
j:,Mins made by the OoC) cliildren were distributed class by class, 
boys and girls separate, but for our discussion a summary of 
this large distribution table must suffice. 'J'his summary appears 
in Table X. 

'IWIilJO X 

Thbi Guumh Gain in Numiiku ok (Jomiiinationh Anhwkiikd ('ourkctlv, 
TBOM Firrv Minutkh of J*itA(,"ncio 



Conil)in;i(i()riH gniiiod . 


-19 

to 


-14 

10 


-9 

to 
5 


-4 
lo 
U 


1 

to 
5 



to 
10 


11 

to 

ir, 


16 

to 
20 


21 

to 
25 


26 

to 
30 


31 

to 
35 


36 

to 

40; 


41 
to 
45 


IndividuiilH 


2 


5 


3 


15 


20 


49 


55 


GO 


G9 


()0 


53 


41 


39 






Per ceuta 


.3 


.8 


.6 


2.5 


3,3 


8.1 


9.1 


9.9 


11.4 


9.9 


8.7 


0.8 


6.4 



TABLE 'X.—Continwd 



Cuiitbinutiuuu gained. 


to 
50 


51 
to 
55 


50 
to 
(50 


Gl 
to 
G5 


G() 
to 
70 


71 
to 
75 


7() 
to 
80 


81 
to 
85 


8G 

to 
90 


91 
to 
95 


to 
100 


Totiil 


IndividualB 


31 


18 


25 


17 


16 


10 


5 


5 


4 


2 


2 


606 






Per centB 


6.1 


3 


4.1 


2.8 


2.6 


1.7 


.8 


.8 


.7 


.3 


.3 


100 







Median 


27.6 combinations 


25 I'ercontilo 


10.7 


75 I'crcoiitilo 


43.4 


iM<;. 


IG 3 


iM':-,.,.,.< A„ 


.t>6 



In I'ablo X the gioss gains are distributed in groups of 5. 
beginning at the end of the tabic one sees that 2 children (or 
.3 per cent of the entire group) worked correctly from 96 to 
100 cond)inations more in the final ten-minute i)eriod than in the 
initial ten-minute i)eriod ; 2 children (or .3 per cent) worked 
from ()i to ()5 more, etc. The distribution is somewhat skewed 



Jmprovcmenl, in the (irouj^ (is a Whole 3 r 

toward the Iii^di end. The range is from a loss of 19 com- 
l)inations to a j^^ain of 100 combinations. 'J'hc median j^ain is 
27.5 combinations. That is, the g'roui) as a whole, measured 
by the median gain, i)rorited by the practice of fifty minutes to 
the extent that it worked correctly in 10 minutes at the end of 
practice 27.5 more cf)mbinations than it worked in 10 minutes 
at the beginning of the ])ractice. This median gain means that 
there were as many children who gained 27.5 combinations or 
more as there were who gained 27.5 combinations or less. The 
U])per 25 ])er cent of the class profited by the i)ractice to the 
extent that they worked c(jrrectly in the final 10 minutes 43.4 
combinations (jr more in excess of the number wf>rked in the 
im'tial 10 minutes. 'i1ie lower 25 ])er cent i)rofited to the extent 
of 10.7 combinations or more, f^nly 25 of the 606 children 
failed to ])rorit by the i)ractice. 

Relative Cain in Division 

What relation did the gross gain from 50 nn'nutes of i)ractice 
bear to the initial ability is our next question for consideration. 
Or, putting it in other terms, what gain per cent in ability to 
make the division associations involved in this experiment re- 
sulted from 50 minutes of practice? This gain for the entire 
grou]) was found from the individual gains made by the 606 
children. 

Table VII, column //, gives these gain per cents for one class. 
P)oy A gained 30 per cent, boy I> gained 78 per cent, boy C, 
168 per cent, etc. The same points that were made in discussing 
the gain ])er cents in addition obtain here. These individual 
gain per cents were distributed class by class, boys and girls 
separate, but only the smnmary of this large table is given in 
Table XI, which shows the form of the distribution, the central 
tendency, and variability of the group. 

In Table XI, the gain ])er cents in division combinations cor- 
rectly answered resulting from 50 mimites of ])ractice are dis- 
tributed in groui)s of 15. The figures show that the distribution 
is skewed t(jward the high end. A wide range, from a loss of 
74 ])er cent to a gain of 400 per cent, is seen. The mode is in 
the grou]) 6t to 75 per cent. The median is 75 ])er cent. That 
is, there were just as many children who gained 75 per cent or 



32 



Practice in the Case of School Children 



more as there were who gained 75 per cent or less. The upper 

25 per cent of the class gained 116 per cent or more, while the 

lower 25 per cent gained 47 per cent or less. The figures in 

the table show that 96 per cent of the children profited by the 

practice. 

TABLE XI 

Gain Pek Cent in Division Combinations Correctly Answered 
FROM Fifty Minutes of Puactice 





-74 


-59 


-44 


-29 


-14 


1 


16 


31 


46 


61 


76 


Gain, per cent .... 


to 


to 


to 


to 


to 


to 


to 


to 


to 


to 


to 




-60 


-45 


-30 


-15 





15 


30 


45 


60 


75 


90 


Individuals 


1 


3 


2 


5 


14 


16 


42 


59 


77 


85 


76 


Per cents 


.2 


.5 


.3 


.8 


2.3 


2.6 


6.9 


9.7 


12.7 


14 


12.5 



Gain, per cent .... 


91 
to 
105 


106 

to 

120 


121 
to 
135 


136 

to 

150 


137 
to 
165 


166 

to 
180 


181 
to 
195 


196 
to 
210 


211 

to 
225 


226 
to 
240 


241 

to 
255 


Individuals 


50 


33 


30 


22 


9 


10 


8 


8 


10 


11 


4 


Per cents 


8.3 


5.4 


5 


3.6 


1.5 


1.7 


1.3 


1.3 


1.7 


1.8 


.7 



Gain, per cent .... 


256 
to 
270 


271 
to 

285 


286 
to 
300 


301 
to 
315 


316 
to 
330 


331 
to 
345 


340 
to 
360 


361 
to 
375 


376 
to 
340 


341 
to 
405 


Total 


Individuals 


4 


5 


2 


2 


3 


2 


3 


4 


2 


4 


606 


Per cents 


.7 


.8 


.3 


.3 


.5 


.3 


.5 


.7 


.3 


.7 


100 



Median 


75 per cent 


25 Percentile 


47 


75 Percentile 


116 


P.E. 


34.5 



Gross Gain in Accuracy 

To know the complete effect of the practice we must measure 
the change in accuracy of making the associations as well as the 
gain in speed in making them. The gain in accuracy is given 
as a gross amount, but is expressed in per cents. Table VII, 
column 18, gives these gross gains for one class. They were 
found by subtracting the numbers in column 4 from the corre- 
sponding numbers in. column 75, Boy A lost 2 per cent, boy 
B gained 4 per cent, etc. The gains made by the 606 children 
were distributed class by class, girls and boys separate, but only 
a summary of the large table is given here, in Table XII, in 



Improvement in the Group as a Whole 



?,2> 



which the form of the distribution, the central tendency and the 
variabiHty of the group are clearly shown. 

TABLE XII 

Gross Gain in Accuracy in Division, Expressed in Per Cents of 
Answers that Were Correct, from Fifty Minutes of Practice 



Per cent gained 


-45 
to 
-41 


-40 
to 
-36 


-35 
to 
-31 


-30 
to 
-26 


-25 

to 
-21 


-20 
to 
-16 


-15 
to 
-11 


-10 
to 
-6 


-5 

to 
-1 




to 




4 


Individuals 


1 


1 





2 


3 


3 


11 


37 


113 


211 


Per cents 


.2 


.2 





.3 


.5 


.5 


1.8 


6.1 


18.6 


34 8 






TABLE 'Xll— Continued 


Per cent gained 


5 

to 
9 


10 
to 
14 


15 

to 
19 


20 
to 
24 


25 

to 
29 


30 
to 
34 


35 

to 
39 


40 
to 
44 


45 

to 
49 


50 
to 




54 


Individuals 


102 


52 


26 


13 


11 


8 


2 


5 


4 


1 


Per cents 


16.8 


8.6 


4.3 


2.1 


1.8 


1.2 


.3 


.8 


.7 


2 







Median 
25 Percentile 
75 Percentile 
P.E. 



2.6 per cent 

1.4 

8. 

4.5 



Table XII shows a distribution of the gross gains in accuracy 
in division expressed in per cents of answers that were correct 
resulting from the 6o minutes of practice. The gains are given 
in groups of 5. The figures show a distribution conforming 
closely to the normal distribution curve. The range is from 45 
per cent lost to 54 per cent gained. The median gain is 2.6 per 
cent which shows that there were as many children who gained 
2.6 per cent or more in accuracy as there were who gained 2.6 
per cent or less. The figures show that 435 children made the 
associations with equal or greater accuracy at the end of the 
practice than at the beginning, while 171 made them with less 
accuracy. 

Combining now the evidence afforded by Tables X, XI, and 
XII, we can say that the 50 minutes of practice effected a very 
remarkable gain in the ability of these children to make the 
division associations concerned in the tests. Not only did the 
group as a whole increase its median ability to the extent that 



24 Practice in the Ca^c of School Children 

in lo minutes at the end of practice it could do correctly 27.5 
(^or 75 per cent) more combinations than in 10 minutes at the 
beginning of the practice, but it also increased by 2.0 per cent 
its median accuracy in giving the answers. 

Summary 

ADDITION 

1. 732 fourth-year children practiced for 75 minutes adding 
such columns as are shown on page 97. The gain from 60 
minutes of the practice was measured. 

2. Initial ability: In the initial 15 minutes of practice the 
median number of columns added correctly was 33.3. 

3. Initial accuracy: In this initial 15-minutc period, the median 
per cent of accuracy was 79 per cent. 

4. Initial al)ility and accuracy: Expressing the initial ability 
by number of jiroblems worked rather than by number of prob- 
lems worked correctly, we can say that in the initial 15 minutes 
the group added a median number of 29.5 with a median accur- 
acy of 79 per cent. 

5. Absolute gain : In the final 15 minutes of practice the group 
added correctly a median of 10.7 more columns than in the initial 
15-minute period. This gain resulted from 60 minutes of 
practice. 

6. Relative gain : This gain of 10.7 columns meant a median 
percentile gain of 48 per cent, from 60 minutes of practice. 

7. Change in accuracy : The change in accuracy was a median 
loss of .4 per cent, which was so small that it hardly needs to 
be considered in measuring the effect of the ])ractice. 

8. The central tendencies given are valid group measures only 
when considered with the wide deviations from them. 

DIVISION 

1. 606 children of last half of third year and first half of 
fourth year practiced for 60 minutes making such division com- 
binations as are shown on page 98. The gain from 50 minutes 
of the practice was measured. 

2. Initial ability: In the initial 10 minutes of practice the 
median number of combinations done correctly was 34.5. 

3. Initial accuracy: In this initial lO-minute period, the median 
per cent of accuracy was 93 per cent. 



Improvement in the Group as a Whole 35 

4. Initial aljility and accuracy : The initial al^lity may be ex- 
pressed in number of combinations worked rather than in number 
worked correctly. The median number of combinations worked 
or attempted in 10 minutes was 37 with a median accuracy of 
93 per cent. 

5. Absolute gain : In the final 10 minutes of practice, the group 
worked correctly a median of 27.5 more combinations than in 
the initial 10 minutes of practice. 

6. Relative gain : This gain of 27.5 combinations meant a 
median percentile gain of 75 per cent from 50 minutes of practice. 

7. Change in accuracy : Along with the median gain of 75 
per cent in speed went a median gain in accuracy of 2.6 per cent. 

TiiFi Value of the Practici-: ILxi'kkimi':nt as a MF/riiOD of 

Teaching 

It would be easier to estimate the value of the practice ex- 
periment as a method for increasing skill if we had norms 
resulting from other methods of work with which to compare 
the results of this study. So far as the author knows, there 
has been no previous attempt to measure in a group of school 
children the progress resulting from the application of any par- 
ticular method of school work. Mr. Courtis measured the 
change in a fifth-year grade from September to June in addi- 
tion and division. However, the material used, the period of 
time for which the change was measured, and the method of 
work were all different, hence a comparison would confuse in- 
stead of clarify. Dr. Thorndike" has made a study of practice 
with university students in addition in which the same method 
and material used in this experiment were used. He found a 
median saving of time of 29 per cent, or improvement of 41 
per cent in amount done per unit of time, the score used here. 
The time spent in practice amounted to about 53 minutes. He 
says : " The amount of improvement in this experiment may 
also add to our confidence that the method of the practice experi- 
ment wherein one works at one's limit and competes with one's 
own past record may well be made a regular feature in many 
school drills. Even if the same length of time produced in chil- 



" Practice in the Case of KcXAiiion, American Journal of Psychology, 
Vol. XXI, pp. 4«3-4««- 



36 Practice in the Case of School Children 

drcn u ])t'iccntilc iiuprovcmcnt only half as j^reat as here, the 
gain would still probably be far greater than the j^ain by any 
of the customary forms of drill." 

This is the nearest norm for a comparison known to the 
author. Dr. Thorndikc's students practiced about 53 minutes; 
the children in the author's e.\])erimcnt practiced 60 minutes or 
about one and a fifth times as lonJ^^ I lis students showed a median 
gain of 41 per cent; the children in this experiment, 48 per 
cent. In 50 minutes of practice in division the children in this 
experiment made a median gain of 75 per cent. 

These gains, taken with Dr. Thorndike's statement that if 
children should make half as j;ieat a gain in the same time as 
was made by the university students in his ex])eriment, they 
would gain more than by methods ordinarily employed in school, 
inspire conlidence in the practice e.\])eriment as a most efiicient 
method for conducting many of the drills necessary in regular 
school work. 

J. C. l>rovvn'^ in a study on the value of drill in arithmetic 
says, " b'ive-minute drill periods upon the fundamental number 
fads, ])receding the daily lesson in arithmetic, were found to 
be beneficial in the sixth, seventh and eighth grades. 
The benefit was not limited to im])rove(l mastery of the number 
habits, but included increased efficiency in arithmetical reason- 
ing. The improvement was still in evidence after the lapse of 
the twelve weeks summer vacation." Air. Rrown does not 
(U'sciibe (lie drill given except as follows: " 'f he first five min- 
utes of each period was devoted to drill work in addition, sub- 
traction, multiplication and division. About one-half of the 
drill work was written and the other half oral." Mr. Hrown 
provided for a control group, lie ftumd that the classes having 
this drill improved from 10 to 20 per cent more in different 
l)hases of arithmetic in the course of 20 recitations than the 
classes that did not have the practice. It is c|uite probable that 
even greater impnnement than was found would have resulted 
had the drill been conducted by the use of practice experiment 
whereby each indiviilual would have been competing with his 
own past record, thus having greater incentive to master thor- 
ouj/liU' the facts involved. 



^Journal of Educalioiuil rsyclio!ot;y, l'\-l>. i()ii, pp. 81 to 88, and Nov. 

1012. pp. 485-492. 



Improvement in the Group as a Whole 37 

Factors Contributing to the Improvement 

That a surprising improvement was made by these children 
working with the practice experiment as a method, has been 
conclusively proved by quantitative data. To try to determine 
some of the factors which contributed to this improvement is 
the present problem, a solution of which must rest upon con- 
clusions more subjective in their nature. But the conclusions 
presented here are products of impressions made on the author 
while observing the reactions of the children to the work of this 
experiment ; or of impressions from previous experience with 
children who studied by different methods the same facts in- 
volved in the experiment; or of impressions of the enthusiasm 
of children engaged in games where a perseverance in under- 
going drudgery was displayed, that is seldom seen in the ])erform- 
ance of school tasks, but whose presence in many phases of 
drill work would guarantee a most efficient and willing mastery 
of the facts involved. Any one who has watched children jump 
the rope, has observed their persistent efforts to outdo a rival, 
or to surpass their own previous performance. Boys will bounce 
a ball from the sidewalk against a wall and catch it with untiring 
zeal so long as there is a contest or a desire to surpass a pre- 
vious record. It is quite noticeable in these games where the 
greatest persistence is displayed that children keep their record 
from time to time. It would be interesting to know if they 
would continue jumping the rope and tossing the balls with any- 
thing near the same enthusiasm if they knew nothing of their 
results from one trial to the next. 

It was with the hope of finding a method for mastering some 
of the fundamental requirements of school work, that would 
at least make a partial appeal to this enthusiastic exuberance in 
children, that the practice experiment was tried, and the author 
feels that this appeal in the method used has been the greatest 
contributing factor to the gains made. However, there are 
many factors which may have entered into the gains made, some 
of which are here presented. 

To what extent was the improvement due : 

1. To increased power of concentrating on the task in hand? 

2. To mastery of the technique of the experiment? 



38 Practice in the Case of School Children 

3. To greater incentive to effort in later tests than in the first 

one? 

4. To work done outside of the time inckided in the test? 

5. The lack of previous use of the functions involved ? 

6. To reinforcement of the mental activity by the motor activi- 

ties involved in writing the results and by the visual per- 
cepts formed? 

7. To maximum of opportunity afforded for exercising the 

function tested? 

8. To the concerted effort of the group, each individual con- 

tributing a share to the gain ? 

9. To acquiring " higher order" habits of work? 

10, To the freedom of the children to regulate their efforts for 

quantity and quality in the way best suited to their indi- 
vidual capacities for each? 

11. To the stimulus resulting from knowing the exact score in 

quantity and quality of the previous test and to the desire 
to surpass this previous record? 

The first five factors, if present to any great extent, would 
tend to produce an exaggerated measure of the gain in adding 
power and dividing power, or the gain found would be made 
up of the improvement in the functions tested plus the gain 
resulting from these five factors. The remaining six factors, 
it seems, would affect the measure of gain only as they directly 
influenced the ability to add and divide. An effort will be made 
to evaluate the influence of each of these eleven factors. To 
what extent was the improvement due to increased power of 
concentrating on the task in hand? In most experiments there 
is an increase in the ability to disregard conditions that make 
against the result sought. The children in this experiment ac- 
quired greater power of concentration and showed greater power 
to withstand distractions. How much this factor influenced can 
not be determined, but whatever the gain was it should be 
regarded as one of the valuable features of the method. 

To what extent was the improvement due to mastering the 
technique of the experiment? In the addition experiment this 
factor certainly did not enter at all, since the children had been 
accustomed to add columns and place their answers beneath 
them. In the division the children had not been accustomed to 
writing short division in the form given on the sheets, but before 



Improvement in the Group as a Whole 39 

the first recorded practice was made they were given enough 
practice in writing the results in this form to insure an under- 
standing of its meaning. For a complete discussion of the care 
that was taken to insure this understanding on their part the 
reader is referred to Chapter I, page 6. There were a few 
children who could not get the results required by these sheets, 
but almost without exception these same children were unable 
to get the results any better by the customary form of division. 
Hence their failure or extreme slowness was due not to the 
form in which the division was expressed, but to their lack of 
knowledge of the process of division. Moreover, such records 
were not used in determining the gain. Further, the class rec- 
ords show no indication of misunderstanding this form of ex- 
pressing division, since there is on the whole as great a gain 
between the performances of any two days as between those 
of the first and second. So the author feels certain that master- 
ing the technique of this experiment produced no greater 
influence on the results, than this same factor ordinarily pro- 
duces in an experiment. 

To what extent was the improvement due to greater incentive 
to effort in later tests than in the first one? The children had 
greater incentive in the later tests because they knew their own 
previous records and were trying to surpass them. They had 
no such record to go against in the first test. Just how much 
this factor contributed to the results can not be told. Every 
other incentive was as great in the first test as in later ones. 

To what extent was the improvement due to work outside of 
the time included in the test? As has been said before, some 
children worked at home to improve. This could not be pre- 
vented. Suggestions not to do so, might have resulted in more 
doing so. Very few did this ; and these had to devise their 
own material, as no sheets were at their disposal. This factor, 
so far as it operated, tended to exaggerate the gains. How much 
can not be judged. While this uncontrolled factor detracts from 
the psychological value of the results, it indicates the value of 
the practice-experiment from the standpoint of education. 

To what extent was the improvement due to lack of previous 
use of the function involved? The amount of improvement that 
may be expected of a group largely depends upon the place it 
has reached in the practice curve. Greater improvement may 



40 Practice in the Case of School Children 

(at least in many cases) be expected in the early stages of the 
curve than in the later stages. One basis is available to show 
the comparative initial ability of the children in this group with 
that of other children in the same school grades. Three fourth- 
year classes containing ii6 children in a school of another system 
using the same addition material worked a median of 21 columns 
with a median accuracy of about 75 per cent in 15 minutes. The 
children in the author's experiment worked a median of 30 
columns with an accuracy of 79 per cent in the initial 15-minute 
period. While the initial ability for the three classes mentioned 
represents but a single school and so can not be considered as 
a norm of ability for fourth-year classes in general, it does indi- 
cate that the children in the present experiment had at least 
a fair degree of ability at the beginning of the practice and that 
there is no good reason for believing that the gains were unduly 
influenced by lack of previous use of the functions tested. 

The five factors thus far discussed would tend in general to 
exaggerate the improvement made to the degree that they oper- 
ated in the group tested. However, the author feels that their 
influence was very slight. The following six factors are the 
ones which best account for the improvement made. 

To what extent was the improvement due to the reinforce- 
ment of the mental activity by the motor activity involved in 
writing the results and by the visual percepts formed of the 
results? This opportunity for improvement was present at 
least in the division to a much greater extent than is true in 
most practice given in school. Formulating the results so that 
they could be written required a complete decision which might 
be shirked in a method not demanding a written answer. Writ- 
ing the results required motor responses which in turn left their 
impression upon the mind. The results when written were ob- 
jective forms to make further lasting impression through the 
sense of sight. The author attributes much of the gain in 
division to the writing of the results which demanded sharper 
thinking, induced motor activity, and made possible more acute 
visual images. 

To what extent was the improvement due to the maximum of 
opportunity afforded by the practice for exercising the function 
tested? Any one who has observed classes drilling by the meth- 
ods commonly used in school knows that much of the time is 



Improvement in the Group as a Whole 41 

consumed in the manipulation of cards, in writing numbers pre- 
paratory for the drill, or in mere dawdling. In the method of 
drill used in the experiments of this study, almost all of the 
energy expended was continually directed toward making the 
associations whose perfecting the drill expected to accomplish. 
There was a minimum of writing to be done, which demanded 
a maximum of thinking, and which also tended to keep the 
mind on the work to be done. One is quite certain that in ten 
minutes of drill with this method, children had the opportunity 
to think and express many more results than with methods 
ordinarily used in school. Greater opportunity for concentrating 
on the facts to be fixed in mind could hardly be devised. So we 
may be certain that much of the gain was due to the children's 
making very many associations within a short time, which pro- 
vided for frequent repetition of associations, one of the funda- 
mental requisites for perfect habit formation. 

To what extent was the improvement due to the concerted 
effort of the group, each individual contributing a share to the 
gain? A prominent element in determining the amount of im- 
provement in any group is the part of the group that contributes 
to the improvement. Effort distributed throughout the group 
gives expectation of greatest improvement. The author has 
never seen children work when there was as great an effort on 
as great a part of a group. 

To what extent was the improvement due to acquiring " higher 
order" habits of work? Very many children in the third and 
fourth years of school form their combinations, especially in 
addition, by counting. No effort was made to get children to 
change their methods. It is not easy to tell with certainty just 
what children use the lower habits and which ones add by use 
of the combinations. No effort was made to keep a record of 
individuals' methods. This could hardly be done because even 
children who are most addicted to adding by ones, add on the 
smaller numbers, like two and three, at one step. So, to make 
an exact classification of the children on this basis would be 
impossible even if one could tell just what method a child were 
using at any instant. However, if children do acquire the higher 
order habits of adding by gradually taking up new combinations, 
it is reasonable to assume that, with the desire these children 
had to advance, many of them acquired higher order habits of 



4 2 Practice in the Case of School Children 

work which contrihutcd a share to the gain that was made. On 
the other hand, this changing to a new method, in some in- 
stances that were very noticeable, caused an increase of errors 
that resulted in a loss. It is believed by the author that many 
of the individual gross losses were due to an unusual increase 
of errors resulting from some such change in method of work. 
Still for the group as a whole it is reasonable to believe that 
some share of the gain was due to the accjuisition of higher order 
habits of work. 

To what extent was the improvement due to the freedom of 
the children to regulate their efforts for quantity and quality in 
the way best suited to their individual capacities for each? In 
most school work such stress is placed on quality of work that 
a child not only is not rewarded for ([uantity of work, but is 
often made uncomfortable for excessive production regardless 
of whether he has maintained his standard of quality. Indi- 
viduals vary greatly in speed and accuracy of work. Rapid 
calculators are as a rule more accurate than slow ones. How- 
ever, this docs not mean that by speeding the slow one up to 
the fast one's norm he would become more accurate ; but there 
is considerable evidence to show that decreasing the speed of 
the fast one to the norm of the slow one would improve his 
accuracy little or not at all. In these tests children were left 
free within certain limits to regulate their effort on (juantity 
and (juality in such a way as to produce the highest score when 
the penalties for inaccuracy had been assessed. Most of the 
children soon saw that they could work more rapidly without 
increasing their errors and regulated their efforts accordingly. 
The fact that the group decreased only .4 of one per cent in 
median accuracy in addition and increased 2.6 per cent in median 
accuracy in division while there was a median gain per cent 
in speed of 50 per cent, shows conclusively that (luantity of 
work may be increased to a vast extent without deterioration in 
its quality. To just what extent the imi)rovcment was due to 
the way the children regulated their effort for quantity and qual- 
ity can not be told, but one may say with assurance that the 
imiM-ovcment would have been much less had the children fol- 
lowed the usual maxim, " Not how much but how well." 

To what extent was the improvement due to the stimulus re- 
sulting from knowing the exact score in quantity and quality 



Improvement in the Group as a Whole 43 

of the previous test and to a desire to surpass this previous 
record? The author feels that the children's knowledge of the 
record of their previous performance, united with the desire each 
one had to surpass his previous record, was the greatest con- 
tributing factor to the improvement. Concerning the value of 
this factor I.add and Woodworth"* say : " Experimental condi- 
tions are stimulating largely because one has a measure of one's 
success and progress ; and the habit of checking up one's work 
can scarcely fail to prove of benefit wherever measures of suc- 
cess and failure are practicable." Not only were the children 
anxious to know the results of their previous tests, but they 
exerted such a determined effort each to surpass his own pre- 
vious record, that one felt progress must necessarily result. At 
the end of a test there were frequent expressions of regret that 
the time was up, but almost no indication of indifference or 
ennui. The children were deriving immediate satisfaction from 
winning in a game where each was striving to be a victor, not 
under the enervating condition that he try to surpass someone 
who is natively his superior, but under the energizing condition 
that he surpass his own previous day's performance. This factor 
provided incentive for the slow child who ordinarily realized 
the futility of striving to do as well as the exceptional members 
of the class ; it provided an incentive also for these excei)tional 
members to exert their best effort, and thus precluded the dawd- 
ling that many methods cultivate among the more gifted members 
of a class. So, under the conditions of this game of practice, 
most of the children were successful, and " nothing succeeds like 
success." Ladd and Woodworth" appraise the emotional tone 
attending successful and baffled effort as follows : " Exactly how 
these emotions act to strengthen one association and to weaken 
or counteract another cannot readily be seen ; but it is safe to 
assume that they corresponded to some genuine dynamic process 
of great efficacy." So, much of the gain in efficiency must have 
been due to the children's knowing their previous success, which 
in turn induced a pleasurable emotional tone to inspire continued 
intense effort and strengthen associations formed. 

The intense effort that was displayed, the improvement that 
was made, and the emotional attitude that was aroused, seem to 

"Elements of Physiological Psychology, p. 571. 
"Ibid., p. 552. 



44 Practice in the Case of School Children 

be features that should recommend the practice experiment as 
a method for school work. IJryan and llarter^" say: "A school 
method must be judged by the moods and tempers which it cul- 
tivates not simply by what is learned, still less by the momentary 
interest it arouses. ]f one forces mastery of the multiplication 
tables by a method which keeps one half the school cowed and 
the other half rebellious, one has obtained a useful result but 
at disastrous cost. Better not know the multiplication tables 
Ih.'iii be thus morally maimed. There are many schools and 
homes where hard tasks are performed in a good temper; where 
thorough (bill does not arrest, but prepares the way for higher 
development, where childien begin to do what they must later 
do to succeed in any business — pass cheerfully from interest in 
desired ends to a resolute dnidgory necessary for the attainment 
of lliese ends." That tlio practice experiment, wherever applic- 
able, meets the demands made here of a method for school work, 
is confidently asserted. 
^^ Psych. Rev., Vol. I, p. 370. 



CiiAi''n':i>: iii 
Till': i<:i'Fi":cT oi^ tiiI': DisrkiiuriMON and length 

OF WORK I'FRiOD Ul'UN Till': KATE 
OF LEARNING 

Tlic proMcin of lliis (li.iplcr is lo iiicasnix- (he cU'vci of ;;iviii;^ 
al)oii( oiu- iiotir of (liill ill arilliiiulic in periods of (IKVcrcnt 
Unt;(li. If we have an iiour lo devole to (h'ill in the division 
lahles. or inulliplicalion tables or in addition, is it better io ^ivc 
this (bill ill short periods extendiii}^ over a eorrespondinjjly 
}j;^realer iiuiiiber of days, or is it better to give the drill in loiif^^er 
periods thns eoneentraliiifj it into fewer days? 

Ti.AN oi" 'III!': rKAc'iicr; 

To understand the maimer in whieb the solution to this prob- 
lem was attempted, it is necessary to recall the plan of j^iving' 
the tests that have been discussed in the prcccdin^^ chapters. In 
the addition experiment with fourth-year children, there were 
four f^roups (;f classes on the basis of the distribution of the 
j)iaeti(e. In all the j^'-roups there was ,an initial practice |)eriod 
and a linal practice-period of fifteen minutes each, but the in- 
terveninj^' forty-live mintites were distributed dilTcrently for the 
different f^roups. In the first j^Moiip (he iiilerveninj;' forty live 
minutes were divided into two ])ractice-perio(ls of twenty-two 
and a half minutes each; in the second ^Mvnip into three practice- 
jjcriods of (ifteeii iiiiiiutcs each; in the third jMoiip into seven 
practice periods of six minutes each and one period of three 
minutes; in the fourth ^n'oup into twenty one two-minute periods 
and one three minute period. The followint^ plan makes this 
distribution clear: 

CinoiirH Initial Picuioi) Intiouvicnino ■1.') MrNnriiH I'^inai, I'lMdoi) 



1 


IT) min. 


2 


22i mill. 






IT) mill 


II 


ir> min. 


.'{ 


15 mill. 






1 Ti mill 


III 


If) mill. 


7 


n mill. 


.•IIKJ 1 


'■'< niin, 


l.'^i mill 



IV l.'> mill. 21 2 mill, iind I '.'< min. I.'i min. 

45 



46 



Practice in the Case of School Children 



Each group of classes practiced once a day on successive 
school days, as far as possible," which of course resulted in 
the experiment extending over a different number of days for 
the different groups. For Group 1, four successive days were 
required; for Group II, five successive days; for Group III, ten 
successive days; and for (n-oup IV, twenty-four successive days. 



TA15LK XI 11 

NUMHKIl OK I'liOHI.lOMS AdDIOI) CoRItlOCTLY IN TIIK INITIAL FlFTEEN- 
MINUTK I'lOUIOl) HY THH FoUll (JUOUPS 

(Each DiaainNATKD by thb Numumk of Minutes in Its Intehvkninq Practich-Periodb) 



Coliiiiuis 


Group I 


Group II 


Group III 


Group IV 










ad.k'd 
CDiTC'ct ly 


22 .^ mill. 


15 min. 


G min. 


2 min. 












Iiidi- 


Per 


Tndi- 


Per 


Iii.U- 


rer 


In.li- 


I'er 




viiluiils 


cents 


viiluiils 


cont.s 


viiliKils 


cent a 


viiiuals 


cents 


to 4 










1 


.5 


2 


.9 


5 to 9 


IG 


8.2 


(') 


5.8 


10 


4.9 


18 


7.9 


10 to 14 


23 


11.9 





8.7 


.35 


17.1 


13 


5.7 


15 to 1!) 


:v2 


10.5 


l(i 


15 I 


45 


22 


40 


17.4 


20 to 24 


;i8 


19. G 


19 


18, :{ 


3(> 


17.0 


37 


10.2 


25 to 2!) 


27 


i;{.9 


11 


10,0 


33 


10. 1 


40 


17.4 


;{0 to ;m 


19 


9.8 


II 


13.5 


19 


9.2 


29 


12.7 


;{5 to ;w 


i;{ 


G.7 


12 


11.5 


11 


5,4 


21 


9.2 


40 to 44 


7 


;i.G 


4 


3,8 


8 


3,9 


14 


0.1 


45 to 49 


•) 


1 


4 


3,8 


4 


• ) 


4 


1,8 


60 to 54 


5 


2,G 


• > 


1,9 


•> 


i. 


4 


1.8 


55 to 59 


4 


2,1 














3 


1.3 


GO to 04 


• ) 


1. 


1 


1. 








1 


.5 


05 to 09 


T 


.5 








1 


.5 


1 


.5 


70 to 74 


4 


2.1 


• > 


1,9 






1 


.5 


75 to 79 
























SO to 84 








;{ 


2 , 9 












85 to 89 


1 


.5 
















90 to 94 














1 


.5 


95 to 99 


















100 to 104 


















105 to 109 






1 


1 










Total 


194 


UK) 


104 


100 


205 


100 


229 


100 


Average 


25.9 


29 . 2 


22.7 


20. 3 


Median 


22.9 


25.4 


21 1 


25 . 1 


25 P. 


IG. 


17.9 


15, 1 


17.5 


75 P. 


82. 


35.8 


28, t) 


33.3 


P.E. 


S. 


8.9 


G.8 


7.9 


•P-1'^t.-oht. Av. 




G 


.9 


.5 


.5 



" For exceptions and further details see Chapter I, " Plan of the 
Practice." 



Distribution and Length of Work Period 47 

In Group T, there were 6 classes, containing 194 children ; in 
Group 11/- 3 classes, containing 104 children; in Group III, 6 
classes, containing 205 children; and in Grouj) IV, 6 classes, 
containing 229 children. 

Addition 
Initial Ability of the Groups 

The classes to compose any group were chosen at random, 
or sometimes on the basis of convenience in reaching them. 
Since nothing was known of the exact ability of the classes in 
addition before beginning the tests, this plan of grouping seemed 
justifiable. Had it been possible to have the tests for all these 
classes in progress at one time, one could have grouped them 
on basis of ability, after the first practice-period, in such a 
way as to secure equality of initial ability in the four groups. 
Table XIII shows a summary of the distribution of the initial 
scores of all these classes grouped, as has been shown above, 
on the basis of the length of the practice-periods for the 45 
minutes of practice between the initial practice-period and the 
final practice-period. The data composing this table are the 
same as those composing Table II, but they are distributed here 
to show the initial ability of the four groups that are being 
compared. The table shows the number of problems worked 
correctly in the initial 15-minute practice-period by the members 
of each group. Samples of the scores distributed here may be 
seen in Table I, column j. 

Table XIII reads as follows : In Group I, 16 children (or 8.2 
per cent of the group) added from 5 to 9 columns correctly in 
the initial 15-minute period; in Group II, 6 children (or 5.8 
per cent of the group) showed the same ability; in Group III, 
10 children (or 4.9 per cent of the group) showed the same 
ability; and in Group IV, 18 children (or 7.9 per cent of the 
group) showed the same ability. The range of the four groups 
was slightly different, but not enough to deserve much weight. 
The central tendencies as shown by the averages of the groups 



"After the experiment had progressed for some time it seemed more 
profitable to discontinue giving the 15-minute practice-periods, since they 
differed so little from the 22^-minute periods, and to concentrate effort 
on the extreme and middle periods. This accounts for the smaller 
number of classes and children in Group II, which in turn causes less 
weight to be given to the measures of Group II in the conclusions. 



48 



Practice in the Case of School Children 



were 25.9, 29.2, 22.7 and 26.3. By medians they were 22.9, 25.4, 
2 1. 1 and 25.1. The individuals comprising the middle 50 per 
cent of the groups had the following ranges: Group I, 16 col- 
umns to 32 columns; Group II, 18 columns to 36 columns; 
Group III, 15 columns to 29 columns; Group IV, 18 columns 
to 32 columns. According to the percentages given in the table, 
about 44 per cent of Group I, 52 per cent of Group II, 38 per 
cent of Group III, and 52 per cent of Group IV, added correctly 
25 columns or more. 

TABLE XIV 

Per Cent of Pkoblems Added Correctly in the Initial Fifteen- 
minute Periods by the Four Groups 

(Each Designated by the Number op Minutrs in Its Intervening Practice-period) 





Group I 


Group II 


Group III 


Group IV 


Per cent, of 
problems 


















22 .^ 


min. 


15 


min. 


6 min. 


2 min. 


correct 




































Indi- 


Per 


Indi- 


Per 


Indi- 


Per 


Indi- 


Per 




viduals 


cents 


viduals 


cents 


viduals 


cents 


viduals 


cents 


6 to 10 


















11 to 15 






1 


1 






1 


.5 


16 to 20 



















21 to 25 


1 


.5 










1 


.5 


26 to 30 


2 


1. 










1 


.5 


31 to 35 


2 


1. 


1 


1. 


2 


1. 


4 


1.8 


36 to 40 


6 


3.1 


1 


1. 


2 


1. 


4 


1.8 


41 to 45 


5 


2.0 


2 


1.9 


5 


2.5 


4 


1.8 


46 to 50 


4 


2.1 


2 


1.9 


7 


3.4 


9 


3.9 


51 to 55 


6 


3.1 








4 


2. 


7 


3.1 


56 to 60 


7 


3.6 


1 


1. 


10 


4.9 


7 


3.1 


61 to 65 


21 


10.9 


5 


4.8 


9 


4.4 


11 


4.8 


66 to 70 


14 


7.2 


7 


0.7 


16 


7.8 


21 


9.2 


71 to 75 


13 


6.7 


17 


16.4 


28 


13.2 


26 


11.3 


76 to 80 


22 


11.4 


12 


11.5 


28 


13.2 


38 


16.7 


81 to 85 


31 


16. 


11 


10.6 


10 


7.8 


23 


10.1 


86 to 90 


27 


13.9 


17 


16.4 


27 


13.2 


37 


16.2 


91 to 95 


19 


9.8 


14 


13.5 


21 


10.3 


22 


9.6 


96 to 100 


14 


7.2 


13 


12.5 


30 


14.6 


13 


5.7 


Total 


194 


100 


104 


100 


205 


100 


229 


100 


Median 




79 




82 




79 




78 


25 P. 




B4 




72 




59 




38 


75 P. 




S8 




91 




c)0 




B8 


P.E. 




12 




9.5 




10.5 




10 



In order to give a more exact statement of the relative initial 
ability of these four groups in addition, it is necessary to show 
the accuracy with which each group did its work. In finding 



Distribution and Length of Work Period 49 

the accuracy of the groups, the same data are used as were 
used in Table III, but now the summaries show the distributions 
for the classes grouped according to the length of the practice- 
periods which comprised the intervening 45 minutes of practice. 
This distribution is found in Table XIV. Samples of the indi- 
vidual records which comprise this table may be found in Table 
I, column 4. 

The medians for accuracy are : for Group I, 79 per cent ; for 
Group II, 82 per cent; for Group III, 79 per cent; and for 
Group IV, 78 per cent. This means that 50 per cent of Group 

1 added with an accuracy of 79 per cent or more; 50 per cent 
of Group II added with an accuracy of 82 per cent or more; 
50 per cent of Group III added with an accuracy of 79 per cent 
or more, etc. 

In terms of number of problems worked and accuracy the 
following is the initial status of each group; Group I added 
29 columns with an accuracy of 79 per cent. Group II added 
31 columns with a median accuracy of 82 per cent. Group III 
added 27 columns with a median accuracy of 79 per cent. Group 
IV added 32 columns with an accuracy of 78 per cent. The 
median initial ability of the groups summarized is as follows: 

Median Number of Median Number of 
Columns Marked Columns Correct Accuracy 

Group I 29 columns 22.9 79 per cent 

Group II 31 columns 25.4 82 per cent 

Group III 27 columns 21.1 79 per cent 

Group IV 32 columns 25.1 78 per cent 

Gross Gain 

Having defined the initial ability of the four groups in terms 
of quality and quantity of performance, we shall now deter- 
mine the absolute gain made by each group in the course of 75 
minutes of practice, 30 minutes of which were occupied by the 
initial 15-minute practice-period and the final 15-minute prac- 
tice-period for all groups alike; the remaining 45 minutes of 
which were divided into practice-periods of 22^/2 minutes for 
Group I, 15 minutes for Group II, 6 minutes for Group III, and 

2 minutes for Group IV. 

The data to be used in reaching a conclusion to this problem 
are the same as those used in Table IV, the source of which 
was described minutely in connection with that table. In Table 



5° 



I'ractkc in the Case of School Children 



XV the gross gains made by the 732 children, samples of which 

may be seen in Table I, column 7./, are distributed for the four 

groups. 

TAliLE XV 

Cliio.sH (Jain in Niimiikii ok Columns Added Couukotly in Fifteen 

MiNii'i'KH, Madio in the Couusio ov Seventy-five 

MiNUTEH OF J'uacti(;e hy the Fouh (Juoui'S 





Group I 


Group II 


Group III 


Group IV 


ColiimiiH 


22i 


min. 


15 min. 


6 min. 


2 min. 
















In.li- 


Por 


Indi- 


Per 


Indi- 


I'or 


Indi- 


Per 




vidiialN 


corit.H 


vidiiulH 


cents 


vid uulH 


cents 


viduals 


cents 


-15 to -12 


2 


1. 


2 


1.9 






1 


.6 


-11 to -8 








4 


3 . H 


5 


2 . 5 


2 


.9 


-7 to -4 


H 


1.1 


4 


3.8 


11 


5.4 


5 


2.3 


~.i to 


12 


(5.2 


H 


7.7 


29 


14.1 


14 


() . 2 


1 to I 


;',5 


J8. 


<» 


S.7 


21 


10.3 


28 


12.2 


r> to s 


;m 


17.5 


10 


9.(5 


28 


13.2 


33 


14.4 


<) to 12 


25 


12.9 


21 


23. 1 


31 


15.1 


31 


13.6 


i;{ to i() 


27 


13. 9 


12 


1 1 . 5 


22 


10.8 


29 


12.7 


17 to 20 


22 


11.3 


7 


(5.7 


25 


12.2 


21 


9.2 


21 (0 21 


4 


2. I 


I 


3 . S 


9 


4.4 


12 


6 4 


25 to 28 


iO 


5.2 


3 


2.9 


7 


3.4 


9 


2.9 


29 to 32 


(> 


3.1 


3 


2 . 9 


9 


4 . 4 


7 


3.1 


33 to M 


(i 


3.1 


3 


2 . 9 


2 


1 


13 


5.7 


37 to 40 


1 


.5 


5 


4.8 


3 


1.5 


7 


3 1 


41 to 44 








1 


J. 


1 


.5 


3 


1.3 


45 to 48 


1 


.5 


2 


1.9 


1 


.5 


3 


1.3 


4«) (0 52 


1 


.5 


J 


1. 








3 


1.3 


53 to 5<) 


















2 


.9 


67 to (iO 












1 


.5 


2 


.9 


61 to (54 






2 


1.9 






2 


.9 


05 to ()8 














2 


.9 


Total 


194 


100 


104 


100 


205 


100 


229 


100 



Average 


11. 


13.0 


10.7 


IG.l 


Median 


9.5 


11. 


9.G 


12.6 


26 P. 


3.5 


4 1 


1.7 


6.4 


76 P. 


17. 


19. 1 


17. G 


23.1 


P.E. 


(5.8 


7.7 


8. 


8.9 


P-E-t.-obt. Av. 


.6 


.8 


.0 


.0 



Tabic XV reads as follows in the bflh lino: " In Group I, 35 
children (or 18 per cent of the group) gained from i to 4 prob- 
lems ; in (Ironp If, <> children (or 8.7 per cent of the group) 
gained fioin 1 to 4 problems; in Group III, 21 children (or 
10.3 per cent of the group) gained from i to 4 problems; in 
Group IV, 28 children (or 12.2 per cent of the group) gained 



Distribution and Length of Work Period 5 1 

from I to 4 problems." Tlic range of distribution is somewhat 
wider for the two-minute grouj) than for the other three groups, 
being from 15 columns lost to 68 gained. The four distribu- 
tions have the same general form, being slightly skewed at the 
high end. The median gain for the 22r/^-minute group is 9.5 
columns; for the 15-minute group, 11 columns; for the 6-minute 
group, 9.6 columns; and for the 2-minute group, 12.6 columns. 
The respective averages were ii.o, 13.6, 10.7 and 16.1. These 
medians and averages show an advantage for the group which 
had the forty-five minutes of practice in 2 minute periods. The 
22^ -minute group and the 6-minute groups made about the 
same gain, while the i5-minute group made a larger gain. The 
percentages in the table show that : 

40 per cent of Group I gained 13 columns or more; 

41 per cent of Group II gained 13 columns or more; 
39 per cent of Group III gained 13 columns or more; 
50 per cent of Group IV gained 13 columns or more. 

Percentile Gain 

To get a measure of the gain of each group in terms of its 
initial performance is our next problem. To find the gain per 
cent of each group of classes, the gain per cents made, by all the 
individuals, samples of which may be seen in Table I, column ij, 
have been distributed in four groups, formed on the basis pre- 
viously described. These data are given in Table XVI. 

Table XVI reads as follows in the seventh line: "In Group 
I, 23 children (or 11.9 per cent of the group) gained from i 
to 15 per cent in columns added correctly; in Group II, 10 chil- 
dren (or 9.6 per cent of the group) gained the same; in Group 
III, 15 children (or 7.5 per cent of the group) gained the same; 
in Group IV, 18 children (or 7.9 per cent of the group) gained 
the same." The range of the groups is from 89 per cent lost 
to more than 300 per cent gained, that of each group being about 
the same. The form of the distribution for each group is simi- 
lar, being skewed toward the high end. 

The median gain per cents for the groups made in the course 
of 75 minutes of practice are as follows : Group I, 45 per cent. 
Group II, 43 i)er cent; Group III, 42 per cent, and Group IV, 
56 per cent. 



52 



Practice in the Case of School Children 



TABLE XVI 



Gain Per Cent in Columns Added Correctly Made in the Course 
OP Seventy-five Minutes Practice by the Four Groups 





Group I 


Group II 


Group III 


Group IV 


Gain, 
per cent 


22^ min. 


15 min. 


6 min. 


2 min. 




Indi- 
viduals 


Per 
cents 


Indi- 
viduals 


Per 

cents 


Indi- 
viduals 


Per 
cents 


Indi- 
viduals 


Per 
cents 


-89 to -75 

-74 to -GO 

-59 to -45 

-44 to -30 

-29 to -15 

-14 to 

1 to 15 

16 to 30 

31 to 45 

46 to 60 

61 to 75 

76 to 90 

91 to 105 

106 to 120 

121 to 135 

136 to 150 

151 to 165 

166 to 180 

181 to 195 

196 to 210 

211 to 225 

226 to 240 

241 to 255 

256 to 270 

271 to 285 

286 to 300 

301 to 305 


1 


2 

4 

11 

23 

29 

28 

29 

24 

15 

8 

5 

5 

3 

2 

1 

1 

1 
1 





1 


.5 


1. 

2.1 
5.7 
11.9 
14.9 
14.2 
14.9 
12.4 
7.8 
4 1 
2.6 
2.6 
1.5 
1. 
.5 
.5 

.5 
.5 




.5 


2 
1 
1 

2 

4 

8 

10 

12 

14 

9 

11 

G 

5 

5 

3 

5 

1 





2 


1 
1 

1 


1.9 

1. 

1. 

1.9 

3.8 

7.7 

9.6 

11.6 

13.4 

8.7 

10.0 

5.8 

4 8 

4.8 

2.9 

4.8 

I 



1.9 



1. 

1. 

1. 


3 

7 

5 

29 

15 

27 

21 

20 

15 

16 

9 

10 

3 

8 

2 

1 
1 
4 
1 
3 


1 
2 
2 


1.5 
3.4 
2.5 
13.7 
7.5 
13.2 
10.3 
9.8 
7.5 
8. 
4.4 
4.9 
1.5 
3.9 
1. 

.5 

.5 
2. 

.5 
1.5 



.5 
1. 
1. 


2 


7 

13 

18 

29 

22 

34 

30 

17 

18 

9 

3 

5 

4 

5 



1 

2 

3 





1 



6 


.9 


3.1 
5.7 
7.9 
12.7 
9.6 
14.9 
12.1 
7.4 
7.9 
3.9 
1.3 
2.3 
1.8 
2.3 


.5 

.9 
1.3 



.5 

2.7 


Total 


194 


100 


104 


100 


205 


100 


229 


100 


Median 
25 P. 
75 P. 
P.E. 


45 
19 
72 
26.5 


43 
13 
86 
26.5 


42 
8 
87 
39.5 


56 
24 
90 
33 



Gain in Accuracy 
One other factor remains to be considered in measuring the 
effect of the practice on the four groups, the change in accuracy 
wrought by the practice. The data to determine this result have 
been used before in Table VI. The source of these data may be 
seen by referring to Table I, column i6. A distribution of the 
gains in accuracy for the four groups is given in Table XVII. 



Distribution and Length of Work Period 



53 



TABLE XVII 

Gross Gain in Accuracy in Addition, Expressed in Per Cents 

OF Answers that Were Correct, R'Iade in the Course of 

Seventy-five Minutes of Practice by the Four Groups 





Group I 


Group II 


Group III 


Group IV 


Per cents 


22* min. 


15 min. 


6 min. 


2 min. 


gained 










Indi- 


Per 


Indi- 


Per 


Indi- j Per 


Indi- 


Per 




viduals 


cents 


viduals 


cents 


viduals 


cents 


viduals 


cents 


-55 to -51 






1 


1. 


2 


1. 


5 


2.3 


-50 to -46 






2 


1.9 












-45 to -41 












2 


1. 






-40 to -36 






2 


1.9 


4 


2. 






-35 to -31 


2 


1. 






12 


5.9 


5 


2.3 


-30 to -26 


2 


1. 


5 


4.8 


5 


2.5 


11 


4.8 


-25 to -21 


3 


1.5 


3 


2.9 


4 


2. 


7 


3.1 


-20 to -16 


9 


4.5 


5 


4.8 


20 


9.8 


20 


8.7 


-15 to -11 


12 


6.2 


7 


6.7 


21 


10.3 


23 


10.1 


-10 to -6 


17 


8.7 


12 


11.6 


19 


9.2 


20 


8.7 


-5 to -1 


23 


11.9 


19 


18.3 


17 


8.2 


42 


18.4 


to 4 


36 


18.5 


19 


18.3 


25 


12.2 


31 


13.6 


5 to 9 


26 


13.4 


8 


7.7 


22 


10.8 


23 


10.1 


10 to 14 


23 


11.9 


7 


6.7 


16 


7.8 


15 


6.6 


15 to 19 


15 


7.8 


6 


5.8 


15 


7.3 


11 


4.8 


20 to 24 


10 


5.2 


3 


2.9 


10 


4.9 


5 


2.3 


25 to 29 


4 


2.1 


3 


2.9 


5 


2.5 


5 


2.3 


30 to 34 


3 


1.5 








2 


1. 


1 


.5 


35 to 39 


5 


2.6 








1 


.5 


2 


.9 


40 to 44 


1 


.5 


1 


1 












45 to 49 


2 


1. 


1 


1. 


2 


1. 






50 to 54 
















1 


.5 


55 to 59 










1 


.5 


2 


.9 


60 to 64 


















65 to 69 


1 


.5 














Total 


194 


100 


104 


100 


205 


100 


229 


100 


Median 


3.5 


-1.6 


-1.5 


-2.7 


25 P. 


-4.7 


-10.1 


-15 


-13.5 


75 P. 


12.9 


6.4 


9.7 


6.2 


P.E. 


8.8 


8.3 


12.4 


9.8 



Table XVII reads as follows near the middle : " In Group 
I, 36 children (or 18.5 per cent of the group) gained from o to 
4 per cent in accuracy; in Group II, 19 children (or 18.3 per 
cent of the group) gained the same; in Group III, 25 children 
(or 12.2 per cent of the group) gained the same; in Group IV, 
31 children (or 13.6 per cent of the group) gained the same." 

The median change in accuracy in the four groups is as fol- 
lows : Group I gained 3.5 per cent; Group II lost 1.6 per cent; 
Group III lost 1.5 per cent; Group IV lost 2.7 per cent. 



54 Practice in the Case of School Children 

Summary 

The following summarized statement gives the data necessary 
for comparing the four groups and for interpreting the results. 

Medians of the Groups of Individuals 



Median 














Initial 














Ability: 


Average 


Median 






Median 


Median 


Examples 


Gross 


Gross 


Reliability: 


Gain 


Gain in 


Correct 


Gain 


Gain 


P.E 


t.-obt. Av. 


Per cent 


Accuracy 


Group I 22 . 9 


11.0 


9.5 




.5 


45 


3.5 


Group II 25.4 


13.6 


11.0 




.8 


43 


-1.6 


Group III 21.1 


10.7 


9.6 




.6 


42 


-1.5 


Group IV 25.1 


16.1 


12.6 




.6 


56 


-2.7 



In computing all averages and medians thus far, the indi- 
vidual in the group has been considered as the unit. We shall 
now present the same data computed, with the class as the unit. 
The median ability of each class was computed ; then the average 
of these class-medians for each group was found. The reliability 
of this average was computed on the basis of the number of 
classes. The following figures show the result. 

Average op Class Medians 





Average Initial 




Average 


Average 




Median Ability 


Reliability 


Gross Median 


Reliability of Median 




of Classes: 


of Average: 


Gain of 


of Average: Percentile 




Examples Correct 


t.-obt. Av, 


Classes 


P.E. ^ . ^ . Gains 
t.-obt. Av. 


Group I 


23.7 


1.7 


10.2 


1.1 42 


Group II 


25.7 


2.3 


9.6 


1.2 41 


Group III 


21.3 


.9 


9.4 


.8 42 


Group IV 


25.5 


1.5 


13.9 


1.9 58 



Using now the best measure of efficiency — the number of 
examples correct, which includes credit for both speed and ac- 
curacy — and the three methods of computing the gain, we have : 





Average 

Gross Gain 

OF Individuals 


Median 

Gross Gain 
OF Individuals 


Average of Median 

Gross Gains 

OF Classes 


Group I 
Group II 
Group III 
Group IV 


11.0 
13.6 
10.7 
16.1 


9.5 
11.0 

9.6 
12.6 


10.2 
9.6 
9.4 

13.9 



It appears that, taking the results at their face value, the 
2-minute practice was the most favorable. Part of its superiority 
to the 225^ -minute and 6-minute practice was probably due to 
the greater initial ability of those taking it. This will be studied 
later. Part of its superiority may have been due also to chance 
factors, but these are as likely to have worked against as for it. 



Distribution and Length of Work Period 55 

Part of its superiority may have been due to the greater length 
of time covered, and so the greater amount of opportunity for 
training in and out of school, apart from the special practice. 
We shall return to this after surveying the facts concerning 
division. 

Division 

Initial Ability 

In the division experiment, with children in the last half of 
the third year and the first half of fourth year, there were three 
groups of classes on the basis of the distribution of the 60 
minutes of practice. In all of the groups there was an initial 
practice-period and a final practice-period of 10 minutes each, 
but the intervening 40 minutes were distributed differently for 
the different groups. In Group I, the intervening 40 minutes 
of practice were divided into 2 practice-periods of 20 minutes 
each ; in Group II, into 4 practice-periods of 10 minutes each ; 
and in Group III, into 20 practice-periods of 2 minutes each. 
The following plan shows clearly this distribution of the practice. 

Initial Intervening 45 Minutes Final 

Groups Practice-Period op Practice Practice-Period 

I 10 min. 2 20 min. periods 10 min. 

II 10 min. 4 10 min. periods 10 min. 

Ill 10 min. 20 2 min. periods 10 min. 

The classes of each of the groups practiced once a day on 
successive school days as far as possible,^^ which of course re- 
sulted in the experiment extending over a different number of 
days for the different groups. For Group I, 4 successive days ; 
for Group II, 6 successive days ; and for Group III, 22 suc- 
cessive days were required to complete the experiment. In 
Group I, there were 6 classes containing 204 children ; in Group 
II, 6 classes containing 209 children ; and in Group III, 6 classes 
containing 193 children. The classes to compose the different 
groups were chosen at random or on the basis of accessibility 
for the author, since nothing definite was known of their ability 
in division before beginning the tests to serve as a basis for 
grouping whereby equality of initial ability might be secured in 
the dift'erent groups. 

Table XVIII shows a distribution of the number of combina- 



^' For exceptions and further details see Chapter I, " Plan of the 
Practice." 



S6 



Practice in the Case of School Children 



tions worked correctly in the initial lO-minute period of prac- 
tice by the 606 children, arranged in three groups made on the 
basis of the length of the intervening periods of practice. Sam- 
ples of these scores may be seen in Table VII, column j. 



TABLE XVIII 

Thk NrTMiiKH OF Division Combinations Answered Cohrectly in 
Tuio Initial Ten-minute Period by Each of the Three Groups 





(irouD I 


( 


jioup II 


G 


roup III 


Numhor of 














combina- 


20 


min. 




10 min. 




2 min. 


tions 
















Iruli- 


Per 


Indi- 




Per 


Indi- 




Per 




viilualH 


cents 


vidui! 


.« 


cents 


vidual 


s 


cents 


5 to 9 


10 


4.9 


10 




4.8 


3 




1.6 


10 to 14 


15 


7.4 


12 




5.7 


6 




3.1 


15 to 19 


19 


9.3 


22 




10.5 


24 




12.4 


20 to 24 


13 


6.4 


24 




11.5 


19 




9.8 


25 to 29 


15 


7.4 


26 




12 . 4 


11 




5.7 


30 to 34 


15 


7.4 


21 




10. 


15 




7.8 


35 to 39 


20 


9.8 


21 




10. 


16 




8.3 


40 to 44 


17 


8.3 


12 




5,7 


1() 




8.3 


45 to 49 


12 


5.9 


19 




9.1 


12 




6.7 


50 to 54 


IS 


s.s 


12 




5 7 


S 




4.1 


55 to 59 


14 


6 . 9 


10 




4.8 


IS 




9.3 


60 to 64 


16 


7.8 


5. 




2.4 


7 




3.6 


65 to 69 


4 


2 


S 




3,8 


:>, 




1.6 


70 to 74 


2 


T. 


3 




1.4 


s 




4.7 


75 to 74 


S 


3.9 


3 




1.4 


5 




2.6 


80 to 84 















3 




1.6 


85 to 89 


2 


1. 


1 




.5 


6 




3.1 


90 1,0 94 


1 


.5 








4 




2.1 


95 to 99 


3 


1.5 








') 




1. 


100 to 101 




















105 to 109 












3 




1.6 


110 to 114 












1 




.5 


115 to 119 




















120 to 124 












1 




.5 


125 to 129 












1 




.5 


Total 


204 


100 


209 


100 


193 


100 


Avenifie 


3 


9.1 




34.6 




44.6 


Median 


;, 


S.3 




32. 




40.3 


25 P. 


9 


2.2 




21.2 




23.5 


75 P. 


5 


4.2 




46.8 




58.3 


P.E. 


1 


6 




12.8 




17.4 


^ •^'•t.-obt. Av. 




1.1 






.9 






1.3 



The average number of combinations worked correctly was : 
for Group I, 39.1; for Group II, 34.6; for Group III, 44.6. 
The median number of combinations worked correctly was : for 



Distribution and Length of Work Period 



57 



Group I, 38.3; for Group II, 32; for Group III, 40.3. The per 
cents in the table show that 47 per cent of Group I, and 35 
per cent of Group II, reached the median of Group III. In 
evaluating the gains made by the three groups, this difference 
in initial ability must be taken into consideration. 

The accuracy of the three groups in doing the division com- 
binations is another factor which must be determined in order 
to show exactly the comparative ability of the groups in their 
initial performance. The data to be used are the per cents of 
accuracy for the 606 children, samples of which may be seen 
in Table III, column 4. These data are distributed in Table XIX 
for the three groups. 

TABLE XIX 

Per Cent of Correct Answers to the Division Combinations in 
THE Initial Ten Minutes by the Three Groups 



Per cent 


Group I 


Group II 


Group III 


of correct 
answers 


20 min. 


10 min. 


2 min. 




Indi- 
viduals 


Per 

cents 


Indi- 
viduals 


Per 

cents 


Indi- Per 
viduals cents 


26 to 30 
31 to 35 
36 to 40 
41 to 45 
46 to 50 
51 to 55 
56 to 60 
61 to 65 
66 to 70 
71 to 75 
76 to 80 
81 to 85 
86 to 90 
91 to 95 
96 to 100 


2 

2 

3 

2 

6 

5 

6 

6 

11 

22 

47 

92 


1. 
1. 

1.5 
1. 

a. 
2.5 
3. 
3. 
5.4 
10.8 

23. 
45.1 


1 

1 
1 
3 
2 

4 
3 
3 
11 
15 
15 
27 
57 
66 


.5 

.5 
.5 

1.4 

1. 

2 

1^4 

1.4 

5.3 

7.2 

7.2 

12.9 

27.2 

31.6 


2 

1 

3 

3 

5 

10 

27 

21 

42 

79 


1. 

.5 

1.6 

1.6 

2.6 

5.2 

14. 

10.9 

21.7 

40.9 


Total 


204 


100 


209 


100 


193 


100 


Median 
25 P. 
75 P. 
P.E. 


94 
87 
98 
5.5 


92 
83 
97 

7 


93 
85 
98 
6.5 



Table XIX reads as follows at the bottom: In Group I, 92 
children (or 45.1 per cent of the group) worked the combina- 
tions in the initial practice-period with an accuracy of 96 to 
100 per cent; in Group II, 66 children (or 31.6 per cent of the 



58 



Practice in ike Case of School Children 



group) worked with the same accuracy; and in Group III, 79 
children (or 40.9 per cent of the group) worked with the same 
accuracy. The three distributions are very similar, being de- 
cidedly skewed toward the low end. The median accuracy for 
the three groups is almost equal. Group I worked with a median 
initial accuracy of 94 per cent, Group II 92 per cent, and Group 
III 93 per cent. So in the factor of accuracy there is not 
enough difference to demand evaluation. 

TABLE XX 

Number of Division Combinations Correctly Answered Gained in the 
Course of Sixty Minutes of Practice by the Three Groups 



Number of 


Group I 


Group II 


Group III 


combina- 
lions 


20 min. 


10 min. 


2 min. 




Indi- 
viduals 


Per 
cents 


Indi- 
viduals 


Per 
cents 


Indi- 
viduals 


Per 

cents 


-19 to -15 
-14 to -10 
-9 to -5 
-4 to 
1 to 5 
6 to 10 
11 to 15 
16 to 20 
21 to 25 
26 to 30 
31 to 35 
36 to 40 
41 to 45 
46 to 50 
51 to 55 
56 to 60 
61 to 65 
66 to 70 
71 to 75 
76 to 80 
81 to 85 
86 to 90 
91 to 95 
96 to 100 


1 
2 



8 

10 

17 

27 

25 

29 

23 

18 

8 

8 

8 

2 

7 

4 

3 

3 





1 


.5 

1. 


3.9 

4.9 

8.3 

13.2 

12.3 

14.2 

11.3 

8.8 

3.9 

3.9 

3.9 

1. 

3.5 
2 

i;5 
1.5 


.5 


1 
3 
2 

6 

5 

25 

18 

29 

26 

20 

19 

13 

15 

12 

3 

2 

4 

4 

2 


.5 
1.4 
1. 

2.8 
2.5 
11.9 
8.6 
13.9 
12.4 
9.5 
9.1 
6.2 
7.2 
5.6 
1.4 
1. 
2. 
2. 
1. 


1 
1 

5 

7 

10 

6 

14 

17 

16 

20 

16 

11 

13 

16 

9 

9 

5 

5 

5 

I 

2 


.5 
.5 
2.6 
3.6 
5.2 
3.1 
7.3 
8.8 
8.3 
10.4 
8.3 
5.7 
6.7 
8.3 
4.7 
4.7 
2.6 
2.6 
2.6 
1.6 
1. 
1. 


Total 


204 


100 


209 


100 


193 


100 


Average 
Median 
25 P. 
75 P. 
P.E. 

^•■^•t.obt.- Av. 


1 

1 


5.1 
2.6 
2.9 
3.5 
0.3 
.7 


1 
1 


5.5 
3.5 
3.4 
6.6 
1.6 
.8 


A 
A 
2 
I 

1 


2.6 
0.4 
6,8 
7.9 
5.6 
1.1 



Distribution and Length of Work Period 



59 



TABLE XXI 

Per Cent of Division Combinations Correctly Answered Gained in the 
Course op Sixty Minutes of Practice by the Three Groups 



Gain, 
per cent 


Group I 


Group II 


Group III 


20 min. 


10 min. 


2 min. 




ImJi- 
\'iduals 


Per 

cents 


Indi- 
viduals 


Per 

cents 


Indi- 
viduals 


Per 

cents 


-74 to -60 

-59 to -45 

-44 to -30 

-29 to -15 

-14 to 

1 to 15 

16 to 30 

31 to 45 

46 to 60 

61 to 75 

76 to 90 

91 to 105 

106 to 120 

121 to 135 

136 to 150 

151 to 165 

166 to 180 

181 to 195 

196 to 210 

211 to 225 

226 to 240 

241 to 255 

256 to 270 

271 to 285 

286 to 300 

301 to 315 

316 to 330 

331 to 345 

346 to 360 

361 to 375 

376 to 390 

391 to 405 


1 

3 

7 

8 

16 

26 

42 

27 

22 

12 

9 

12 

4 

6 

1 

1 

1 

1 

1 

1 

1 

1 

1 


.5 

1.5 

3.5 

3.9 

7.8 

12.8 

20.6 

13.2 

10.8 

5.9 

4.4 

6. 

2. 

3. 

.5 

.5 

.5 

.5 

.5 

.5 

.5 

.5 
.5 


3 

2 

2 

5 

5 

18 

24 

20 

31 

27 

14 

14 

6 

8 



3 

5 

3 

3 

8 

1 

1 



1 





1 

1 
1 

2 


1.4 
1. 
1. 

2.5 

2.5 

8.6 

11.2 

9.6 

14.8 

12.9 

6.6 

6.6 

2.8 

3.8 



1.4 
2.5 
1.4 
1.4 
3.8 
.5 
.5 

.5 

.5 
.5 
.5 

1. 


2 

3 

8 

9 

15 

27 

27 

24 

10 

12 

10 

3 

6 

2 

4 

6 

2 

2 

3 

5 

2 

1 

2 

1 

2 

2 

2 

1 


1. 

1.6 
4.1 
4.7 
7.8 

14. 

14. 

12.5 
5.2 
6.2 
5.2 
1.6 
3.1 
1. 

2.1 
3.1 
1. 
1. 
1.6 
2.6 
1. 

.6 
1. 

.5 
1. 
1. 
1. 
.5 


Total 


204 


100 


209 


100 


193 


100 


Median 
25 P. 
75 P. 
P.E. 

P-^-t.-obt. Av. 


60 
40 
92 
26 

1.8 


73 
41 
112 
35.3 
2.4 


94 

67 

147 

40 

2.9 



Gross Gain 

To determine the gross gain of the three groups in division 
made in the course of 6o minutes of practice, the gross gains 
in number of combinations worked correctly by each of the 



6o Practice in the Case of School Children 

606 children, samples of which may be seen in Table VII, 
column 16, were distributed for the three groups in Table XX, 
and the measures for each group calculated. 

The average number of combinations gained in the course of 
60 minutes by Group I was 25.1; by Group II, 25.5; and by 
Group III, 42.6. The corresponding medians were 22.6, 23.5 
and 40.4, At their face value these facts show a very decided 
advantage for the group which did the intervening 45 minutes 
of practice in 2-minute periods over the groups which did it 
in lo-minute periods and 20-minute periods. Only 18 per cent 
of the 20-minute group and 21 per cent of the lo-minute group 
reached the median of the 2-minute group. 

Percentile Gain 

The data used in determining the gain per cent for the groups 
are the individual records of the 606 children in gain per cent, 
samples of which may be seen in Table VII, column 17. These 
gain per cents are distributed (separately) in Table XXI, for 
the three groups. 

Gain in Accuracy 

The gross gains in accuracy for the 606 individuals, samples 
of which may be seen in Table VII, column 18, were distributed 
for the three groups in Table XXII and the measures for each 
group found. 

The median gains in accuracy were: for Group I, 2.1 per cent; 
for Group II, 3.5 per cent; for Group III, 2.3 per cent. These 
gains are so nearly the same that we can say that in division 
no advantage was shown in accuracy from one length of practice 
period over the others. All periods resulted in about an equal 
absolute gain in accuracy. 

Sitmmary 

There were 18 3B and 4A classes, containing 606 children 
in the division experiment. Group I consisted of 6 classes con- 
taining 204 children ; Group II consisted of 6 classes containing 
209 children; and Group III consisted of 6 classes containing 
193 children. The groups were made on the basis of length of 
practice-period in the 40 minutes of practice between the initial 
lo-minute period and the final lo-minute period, which were the 



Distribution and Length of Work Period 



6i 



TABLE XXII 

Gross Gain in Accuracy in Division, Expressed in Per Cents op 
Answers that were Correct, Made in the Course of Sixty 
Minutes of Practice by the Groups 





Group I 


Group II 


Group III 


Gain, 
per cent 


20 min. 


10 min. 


2 min. 




Indi- 
viduals 


Per 

cents 


Indi- 
viduals 


Per 

cents 


Indi- 
viduals 


Per 

cents 


-45 to -41 

-40 to -36 

-35 to -31 

-30 to -26 

-25 to -21 

-30 to -16 

-15 to -11 

-10 to - 6 

- 5 to - 1 

Oto 4 

5 to 9 

10 to 14 

15 to 19 

20 to 24 

25 to 29 

30 to 34 

35 to 39 

40 to 44 

45 to 49 

50 to 54 


1 

1 

6 
15 
36 

82 
28 
17 
3 
4 
5 
4 
2 


.5 

.5 

.3 

7.4 
17.6 
40.2 
13.7 

8.3 

1.5 

2 

2.5 

2 

1 


1 

2 

1 

1 

6 

38 

70 

43 

13 

10 

8 

5 

3 

4 

4 


.5 

1 
.5 

.5 

2.8 

18.1 

33.5 

20.6 

6.2 

4.8 

3.8 

2.4 

1.4 

2 
2 


1 

3 

4 

16 

39 

59 

31 

22 

13 

1 

1 

1 

1 

1 


.5 

1.6 

2.1 

8.3 

20.2 

30.6 

16.1 

11.4 

6.7 

.5 

.5 

.5 

.5 

.5 


Total 


204 


100 


209 


100 


193 


100 


Median 
25 Percentile 
75 Percentile 
P.E. 


2.1 

-1.6 

G.6 

4.2 


3.5 
-.3 
8.9 
4.6 


2.3 
-2.4 

8.2 
5.3 



same for all three groups. In Group I, these intervening practice 
periods were 20 minutes each; in Group II, 10 minutes each; in 
Group III, 2 minutes each. The following is a summary of the 
important facts concerning the differences in improvement. 



Group I 
Group II 
Group III 



Medians of the Groups op Individuals 



Median 
Initial 
Ability 

38.3 

32. 

40.3 



Average 
Gross 
Gain 



25.1 
25.5 
42.6 



Median 
Gross 
Gain 



22,6 
23.5 
40.4 



Reliability 
of Average 

t.-obt. Av. 



1.1 



Median Median 

Gain Gain in 

Per cent Accuracy 



60 
73 
94 



2.1 
3.5 
2.3 



62 Practice in the Case of School Children 

1. The initial ability of Group II was enough below that of 
the other two groups to demand careful consideration in evalu- 
ating the relative gain. 

2. The initial accuracy of the groups was so nearly equal 
that it may be considered a constant factor in the discussion. 

3. Group III gained almost twice as many combinations done 
correctly in the course of the 60 minutes of practice as did the 
other two groups, whose gains were practically the same. 

4. In gross gain in accuracy the three groups were so nearly 
equal that this too may be considered as almost a constant factor. 

The measures given have been computed from the scores of 
the individuals comprising these groups. The following measures 
are computed from the scores of the classes in the groups. The 
median for each class was found, then the average of these 
medians was computed. 

Average op Class Medians 

AvernKe _ _ Average Average 

Initial Median Reliability Gross Reliability of Medinn 

Ability of of Average: Median of of Average: Percentile 

Classes P.E. . , . . Classes P.E. . , , Gains 

t.-obt Av. t.-obt. av. 

Group I 38.4 2.2 20.6 2. 58 

Group II 33.4 1.7 25.1 1.5 77 

Group III 41.4 4.5 44.7 2.7 114 

Using the three methods of computing gross gain, we have: 

Average Median Average op Median 

Gross Gain Gross Gain Gross Gains 

OP Individuals op Individuals op Classes 

Group I 25.1 22.6 20.6 

Group II 25.5 23.5 25.1 

Group III 42.6 40.4 44.7 

Considering the facts for both addition and division, it appears 
that, subject to discounts for the inequalities of the groups in 
initial ability, there is considerable advantage in the short period 
lengths, when the length is two minutes. The advantage there is 
noteworthy, since in addition the gain is greater than in the 
longer-period groups even when their ability was greater in 
the longer period (Group II) ; and since in division the gain is so 
very much greater than in the 20- or lo-minute group. 

The facts can be freed from the influence of inequalities in 
ability at the beginning of practice by comparing only those of 



Distribution and Length of Work Period 63 

equal initial ability. For example, we find in the case of division 
that those of initial ability 15 averaged in gain, 18.0, 26.3 and 
23.0 according as they had practiced in 20-, 10-, or 2-minute 
periods; those of initial ability 16 averaged in gain 16.4, 15.3 and 
41.0 according as they had practiced in 20-, 10-, or 2-minute 
periods. 

Making such calculations for those of each initial ability in 
division from 5 to 64 and allowing equal weight to each succes- 
sive set of five successive groups, it appears that on the average 
the 20-minute, lo-minute, and 2-minute period varieties of prac- 
tice brought to those of equal initial ability gains in the relation 
of 100, 1105^ and 177. 

In the case of addition the same procedure, carried out with 
those of initial abilities 5, 6, 7, and so on up through 49 gives 
the following results : According as the practice was in 22^ , 
15, 6, or 2-minute divisions, it brought to those of equal initial 
ability gains in the relation of 100, 121, loi and 146^. 

It appears, then, that the superiority of the shortest practice- 
period length remains when inequalities of initial ability are 
eliminated. It appears further that the periods of intermediate 
length have really a greater superiority over the longest period 
than the results irrespective of differences in initial ability 
showed. There is a positive relation between initial ability and 
gross gain. Consequently Group III in addition and Group II 
in division, which happened to be groups of low initial ability, 
suffered in the comparison. 

The detailed facts for the gains of each variety of initial 
ability according to the nature of the practice are shown in 
Tables XXIII and XXIV. 



64 



Practice in the Case of School Children 



TABLE XXIII 

The Relation of Length op Practice-Periods to Gross Gain 
IN Addition 



Initial 


22^ min. 


15 min. 


6 min. 


2 min. 


ability. 
Exam- 


practice- 


periods 


practice- 


periods 


practice- 


periods 


practice- 
Average 

Gross 
Gain 


periods 


ples 
correct 


Average 
Gross 
Gain 


Num- 
ber of 
Cases 


Average 
Gross 
Gain 


Num- 
ber of 
Cases 


Average 
Gross 
Gain 


Num- 
ber of 
Cases 


Num- 
ber of 
Cases 



1 
2 
3 














2 


1 


4 


5 


1 






14 


1 






5 


4 


2 


8 


2 


4 


1 


4 


1 


6 














30 


4 


7 


9.3 


3 


8. 


2 


2.5 


2 


14 2 


4 


8 


7.6 


3 


9.5 


2 


11.2 


5 


9.8 


5 


9 


13.4 


7 






-1.5 


2 


4.7 


4 


10 


6.6 


3 


25 


1 


14.7 


11 


13.3 


3 


11 


19 


1 


2 


2 


13 


4 






12 


7 


9 


4 


3 


17.7 


4 


9.2 


4 


13 


10.5 


4 






9.3 


6 


8.8 


5 


14 


7.3 


6 


11 


3 


12.8 


8 


23.3 


3 


15 


7.6 


3 


-6 


2 


14.6 


9 


19.1 


12 


16 


5.2 


6 


21.3 


3 


6.2 


5 


16.7 


8 


17 


7.5 


9 


5 


3 


5 


8 


14.2 


4 


18 


9.3 


3 


13.3 


4 


9 


21 


8 


9 


19 


7.6 


11 


15 


3 


4 


5 


8.2 


8 


20 


7.5 


10 


6.1 


7 


10.8 


16 


11.1 


7 


21 


9.7 


9 


10.2 


5 


6 


2 


17.8 


6 


22 


6.7 


4 


7 


1 


12.7 


4 


9.4 


12 


23 


6.3 


7 


14 


2 


6.9 


10 


4.3 


7 


24 


12.6 


8 


7.7 


4. 


9.6 


5 


20.8 


6 


25 


10 


6 


21 


2 


8.1 


8 


25.8 


5 


26 


7.2 


4 


28.3 


3 


7.3 


7 


11.5 


11 


27 


8.6 


8 


9.6 


3 


10.6 


7 


17.2 


5 


28 


7.5 


6 


9 


1 


8.6 


8 


15 


13 


29 


10.8 


4 


-3 


1 




2 


11.3 


6 


30 


23.7 


6 


8.4 


5 


1.4 


5 


10.3 


4 


31 


12.4 


5 


16.3 


4 


6.8 


4 


11.9 


10 


32 


14.5 


2 


22.6 


3 


3.5 


2 


25.4 


5 


33 


15 


3 


37 


1 


16 


3 


27.3 


6 


34 


5.6 


3 


2 


1 


14.4 


5 


11.8 


4 


35 


4.8 


4 


3 


1 


13.4 


7 


14.3 


3 


36 


11.8 


4 


8.8 


4 


19 


1 


34.5 


2 


37 






12.8 


5 


29 


1 


10.6 


8 


38 


8 


1 


14 


1 


22 


2 


17.6 


5 



Distribution and Length of Work Period 
TABLE XXIII— Continued 



6S 



Initial 


22* min. 


15 min. 


6 min. 


2 min. 


ability. 
Exam- 


practice- 


periods 


practice- 


periods 


practice- 


periods 


practice- 


periods 


ples 
correct 


















Average 
Gross 
Gain 


Num- 
ber of 
Cases 


Average 
Gross 
Gain 


Num- 
ber of 
Cases 


Average 

Gross 
Gain 


Num- 
ber of 
Cases 


Average 
Gross 
Gain 


Num- 
ber of 
Cases 


39 


20 


3 


61 


1 






35 


2 


40 


15 


1 


24 


1 


10.7 


6 


33.6 


5 


41 


7 


1 


3 


1 






38 


3 


42 


23.5 


2 


21.5 


2 


37 


1 


28.5 


2 


43 


20 


1 










15 


3 


44 


4.5 


2 






28 


1 


8 


1 


45 


27 


1 










16 


1 


46 


23 


1 


14 


1 










47 










16.5 


2 






48 


33 


1 






26.5 


2 






49 






49 


2 






26.3 


3 


50 


26 


1 


4 


1 






34 


3 


51 


39 


1 


-6 


1 


19 


1 






52 


30 5 


2 


18 


1 










53 


30 


1 






29.5 


2 






54 














53 


1 


55 


















56 














57 


1 


57 


29 


1 














58 














59 


1 


59 


17.5 


2 










33 


1 


60- 64 


20.5 


2 


31 


1. 






12 


1 


65- 69 


28 


1 






46 


1 


24 


1 


70- 74 


30.3 


3 


37 


2 






43 


1 


80- 84 






23 


3 










85- 89 


22 


1 














90- 94 


IS 


1 










18 


1 


105-109 






48 


1 










Total 




194 




102 




207 




231 



66 



Practice in the Case of School Children 



TABLE XXIV 

The Relation of Length of PnACTiCE-PERiODS to Gross Gain 
IN Division 



Initial 
ability. 


20 min. 


10 min. 


2 min. 


practice-periods 


practice-periods 


practice-periods 


Examples 
correct 














Average 


Number 


Averaee 


Number 


Averocte 


Number 




Gross Gain 


of Cases 


Gross Gain 


of Case.s 


Gross Gain 


of Cases 


5 


20 


1 


15.5 


2 






6 


13 


2 


12 


3 






7 


9 


1 


10.5 


2 






8 


15.3 


3 


5 


1 


24 


1 


9 


4.6 


3 


11.5 


2 


12 


2 


10 


10 


3 


8 


1 






11 


9.6 


3 


21 


4 






12 


12 


2 


5 


3 


24 


2 


13 


13.8 


5 


13.5 


2 


35.5 


2 


14 


39 


2 


14.5 


2 


38 


2 


15 


18 


4 


26.3 


4 


23 


7 


16 


16.4 


5 


15.3 


3 


41 


4 


17 


15.5 


6 


13 


4 


38.2 


5 


18 


2 5 


2 


40.5 


6 


44.7 


6 


19 


15 


2 


11.8 


5 


41 


1 


20 


-15 


1 


26.2 


9 


34.5 


2 


21 


28 


2 


38.5 


2 


38.8 


5 


22 


23 


2 


27.5 


8 


43.6 


5 


23 


30,6 


3 


20.5 


4 


17.7 


3 


24 


14.2 


5 


30.5 


4 


74.7 


3 


25 


21.3 


3 


21.2 


6 


57 


1 


26 


25 


1 


13 


2 


40.2 


5 


27 


10.5 


4 


17.1 


7 


20.5 


2 


28 


22.3 


4 


25 4 


s 


23 . 5 


2 


29 


25.3 


3 


41.5 


2 


19.5 


2 


30 


23.6 


5 


19.5 


6 


38.4 


5 


31 


18.5 


4 


20 


4 


40.5 


2 


32 


16.5 


2 


28.6 


3 


33.5 


2 


33 


28 


2 


31.6 


3 


39.3 


4 


34 


23 


2 


29 


4 


33.5 


2 


35 


21 


3 






44 


5 


36 


19.7 


7 


31.8 


5 


44 


1 


37 


22.3 


4 


31.1 


8 


46.8 


4 


38 






27.3 


3 


48.5 


5 


39 


25.3 


6 


20.8 


5 


17 


2 


40 


19 


4 


7.5 


2 


43 


1 


41 


46 


1 


30.2 


5 


46.3 


3 


42 


32 


2 






39.3 


4 


43 


25.4 


8 






34 


8 


44 


22.5 


2 


13 


5 


45 


1 



Distribution and Length of Work Period 87 

TABLE XXIY— Continued 



Initial 


20 min. 


10 min. 


2 min. 


ability. 


practice- 


periods 


practice- 


;)eriods 


practice-periods 


Examples 
correct 














Average 


Number 


Average 


Number 


Average 


Number 




Gross Gain 


of Case.s 


Gross Gain 


of Cases 


Gross Gain 


of Cases 


45 


10.5 


2 


18 


4 


64 


1 


46 


6 


1 


38.3 


6 


18 


3 


47 


26 


1 


16 6 


3 


41.6 


3 


48 


32 


4 


15.5 


2 


47.5 


2 


49 


37.5 


4 


25.8 


4 


44.3 


4 


50 


17.5 


2 


44 


1 


27 


1 


51 


16,6 


7 


29 


7 


42.8 


4 


52 


24.3 


3 


6 


1 


37.5 


2 


53 


32 


3 










54 


28.3 


3 


22 


3 


67 


1 


55 


19.5 


4 


19.5 


2 


31.8 


6 


56 






49 


1 


44.5 


2 


57 


37 


4 


27.8 


6 


40.5 


2 


58 


18.3 


3 


50 


1 


27.8 


4 


59 


23.6 


3 






37.3 


3 


60 


26 


2 


27 


1 


83 


1 


61 


25 


3 


47 


1 


45.3 


3 


62 


31 


3 


34 


3 






63 


41.4 


5 


49 


1 


47 


1 


64 


42 


3 






48 


1 


65 


51 


1 


63 


1 


62 


1 


66 


35 


1 


48.5 


2 


92 


1 


67 


31 


1 


48 


1 






68 


40 


1 


34 . 5 


2 


43 


1 


69 














70- 74 


69.5 


2 


28 . S 


4 


57.9 


9 


75- 79 


45.4 


8 


46.6 


3 


49.6 


5 


80- 84 










63 


3 


85- 89 


64 


2 


41 


1 


43 


6 


90- 94 


90 


1 






74 


4 


95- 99 


59.3 


3 






77.5 


2 


100-104 


67 


1 










105-109 










87 


2 


110-114 










62 


1 


115-119 














120-124 










90 


1 


125-129 










72 


1 


Total 




205 




210 




192 



68 



Practice in the Case of School Children 




Distribution and Length of Work Period 69 

General Summary 

While the results of both the addition and division experiment 
show that the greatest gains were made by the groups which 
had their practice in the shortest periods, we can not conclude 
that all of this excess of gain was due to the difference in the 
length of the practice-period. Other factors probably contributed 
a share to this extra gain, (i) The groups, working in shorter 
periods, because of the number of days over which the experi- 
ments ran, had greater opportunity during the experiment to 
profit from the regular school work than other classes. This 
difference might have been overcome by giving the experiment to 
all groups in the same number of days, which would necessitate 
giving more than one of the shorter practice-periods within a day 
or allowing days to lapse between the longer practice-periods. 
This is a problem for another study. (2) The groups working in 
shorter periods had a longer time in which to catch the spirit of 
the experiment and to become enthusiastic over surpassing their 
previous performance. They had their records read to them 
more times and had the incentives to intense effort repeated more 
often. (3) They also had greater opportunity and incentive to 
do work outside of the time given to the experiment. 

On the other hand the 2-minute group in addition was handi- 
capped by the difficulty of making an absolute gain from one 
2-minute period to the next. The median gross gain for this group 
was 12.6 columns from 23 periods of practice which shows that 
only those who gained most rapidly could have had the pleasur- 
able effect from day to day of surpassing their previous day's 
record. Then, too, if they did surpass their previous days record 
it was by a small amount which did not give the same feeling of 
success and incentive to effort that a larger gain would have 
given. This condition did not operate much in the 2-minute 
period in division where the median gain was 40.4 combinations 
in 20 periods of practice. In all other periods of practice the gain 
from day to day was larger and resulted in a more intense 
feeling of success and greater incentive to effort. This handicap 
on the 2-minute group, it seems, would offset the extra gains 
from conditions previously mentioned. 

From the administrative side of school-room work, the longer 
periods of practice can be much more economically managed in 



70 Practice in the Case of School Children 

that preparation needs be made for them much less often. Per- 
haps this factor would compensate in a large measure for the 
disadvantage which the long period seems to have in comparison 
with short ones. 



CHAPTER IV 
THE PERMANENCE OF THE PRACTICE EFFECT 

In order to measure the permanence of the abiHty acquired in 
the course of the experiments that have been described, retention 
tests were given at the end of the school year in June, 1912, 
to those classes that had taken their practice during the preceding 
four months and finished it long enough before the end of June 
to make a retention test seem profitable. 

No one knew that these later tests were to be given after the 
regular experiment was finished. This makes it reasonable to 
suppose that about the regular amount of work was done in 
addition and division from the close of the experiment until the 
tests for retention were given and that no extra drill was given 
in anticipation of them. These tests afford data by which to 
determine to what extent the efficiency acquired during the experi- 
ments persisted from the close of the practice in the experiment 
to the end of June under the normal exercise of performing 
regular school work. 

Retention tests were also given at the beginning of the follow- 
ing school year in September to as many of the same classes as 
the author had access to. These tests afford data by which to 
determine to what extent the efficiency shown at the end of June 
persisted through vacation, during which practice was substan- 
tially zero. Immediately following these retention tests in 
September, the classes were given sufficient practice to restore 
them approximately to the same efficiency they had attained at 
the end of their practice the preceding year. 

These retention tests correspond in every way to the initial 
and final practice-periods of the experiment. In addition there 
were practice-periods of 15 minutes, and in division practice- 
periods of 10 minutes. Hence, the scores made in them were 
directly comparable to the scores in both the initial and final 
periods of the experiment. 

In the practice conducted to restore the ability lost during 

71 



7 a Practice in the Case of Scliool Children 

vacation some fivc-niinutc practice-periods were used, but the 
last ])racl ice-period jj;iven was a 15-iiiinute ])eriod in addition 
and a lo-niiiuite period in division. The retention tests and the 
practice f^iven in Septciuljer were conducted by the author in 
exactly the same manner as the practice of the experiment had 
been conducted. During some of the 15-minute tests in 
September some of the children showed signs of weariness that 
were not noticeable during the original practice. The author 
flit that this was due to the children's inability to apply them- 
selves to mental work immediately after vacation with the same 
jiersistence that they had shown during the experiment, which 
was conducted at a time when they were accustomed to do con- 
tinuous mental work. There is no doubt in the author's mind 
that shorter periods of practice would have been more effective 
in this practice given during the first two weeks of school, and 
that the time required for these classes to regain their former 
cfliciency would have been considerably lessened if given in 
shorter periods. The present experiment affords no data to 
prove this opinion, but it would be i)rofitable to iind out if 
practice-periods should be shorter in the early weeks of the 
school term than in the later ones. 

This chapter treats, then, three phases of the problem of the 
permanence of associations: (i) The permanence of well- 
practiced addition and division associations during a period of 
time when they were normally used in regular school work. 
(2) The permanence of the same associations after almost com- 
plete disuse during the summer vacation. (3) The permanence 
of these associations as shown by the amount of practice required 
to restore them to their former efficiency. 



Pl'-RMANKNt'l': OK ASSOCIATIONS NORMALLY ITsiCD IN SciIOOL 

Work 
AnniTioN 

Nine classes that had taken the addition practice finished the 
experiment and took the retention test in June at the time indi- 
cated below : 



The Permanence of the Practice Effect 73 



Class 


Final Pkactice Teuiou 


Retention Teht 


XIV 


March 28 


June 21 


XVI 


March 28 


June 20 


XIII 


March 2U 


Juno 21 


XVII 


April 18 


June 20 


VIII 


April I'J 


June 19 


IX 


April I'J 


June 18 


X 


May 9 


June 19 


XI 


May 9 


Juno 19 


V 


May 27 


June 20 



The question which wc are considering is, What change in 
ability to add took place in the children of these nine classes in 
the time intervening between the final practice-period and the 
retention test near the end of June, a time varying for the 
different classes from 12 weeks to 3 weeks, in which no si)ecial 
drill was given in arithmetic, but in which regular work in 
arithmetic was in progress? The scores made in the retention 
tests were not compared directly with the scores made in the 
final i)ractice-period of the experiment, but with the scores made 
in the initial practice-period. So, to answer the question ])ro- 
posed above, we shall first find what gain was made in the reten- 
tion test near the end of June over the initial test and then 
compare this gain with that made during the original practice- 
experiment by these same classes. Any difference found l)etween 
these two gains will be the change that took place in the interval 
between the close of practice and the test near the end of June. 

Gross Gain: Table XXV gives under (a) the gross gain of 
these nine classes in the retention test at the end of June over 
the initial period of the experiment and under (b) the gross 
gain of these same classes during the original experiment. The 
median gain in columns added correctly in the retention test 
near the end of June over the initial ])ractice-i)eriod of the 
experiment was 11 columns, while the median gain of these 
same classes during the experiment was 10.4 columns. This 
means that, in the time elapsing between the end of the regular 
practice of the experiment and the end of June in which only 
the regular school work in addition was carried on, these classes 
not only did not lose any of the ability acquired during the 
experiment, but gained the difference between these two medians 
or .6 column. The 25 percentile and the 75 percentile together 
with the median indicate that this gain was distributed all along 
the curve. 

Gaifi in Accuracy: Table XXVI gives under (a) the gain of 



74 



Practice in the Case of School Children 



these nine classes in accuracy in the retention test at the end of 
June over the initial period of the experiment, and under (b) 
the gain of the same classes in accuracy during the experiment. 
Between the initial practice-period and the retention test there 
was a gross gain in accuracy of .2 per cent while during the 
experiment there was a loss in accuracy of .4 per cent. The 
difference between these two medians, .6 per cent, is the median 
gain in accuracy from the end of the original practice to the end 
of June. This indicates a slight gain in accuracy of performance 
after the intense practice ceased and normal use of the associa- 
tions was continued. However, this gain is so small that little 
weight should be given to it. 

TABLE XXV 



(a) 
Gross Gain in Addition 
FROM THE Initial Period 
OF THE Experiment to 
THE End of June 



(b) 
Gross Gain of the Same 
Individuals During the 
Experiment 



Columns 
gained 


Individuals 


Per cents 


Individuals 


Per cents 


-15 to-12 


4 


1.6 


1 


.4 


-11 to - 8 


3 


1.5 


6 


2.6 


- 7 to - 4 


7 


2.7 


7 


2.7 


- 3 to 


14 


5.4 


25 


9.7 


1 to 4 


35 


13.5 


40 


15.5 


5 to 8 


39 


15.1 


35 


13.5 


9 to 12 


44 


17. 


31 


11.9 


13 to 16 


38 


14.3 


34 


13.1 


17 to 20 


24 


9.3 


29 


11.2 


21 to 24 


13 


5.1 


14 


5.4 


25 to 28 


4 


1.6 


13 


5.1 


29 to 32 


12 


4.7 


9 


3.5 


33 to 36 


5 


1.9 


6 


2.6 


37 to 40 


4 


1.6 


1 


.4 


41 to 44 


6 


2.6 


1 


.4 


45 to 48 


2 


.8 


3 


1.5 


49 to 52 






1 


.4 


53 to 56 


1 


.4 






57 to 60 






1 


.4 


61 to 64 


1 


.4 


1 


.4 


65 to 68 


2 


.8 






Total 


258 


100. 


258 


100. 




Median 


11 


Median 10.4 




25 Percentile 


4.6 


25 Percentile 3 . 1 




75 Percentile 


18.1 


75 Percentile 17.1 




P.E. 


6.8 


P.E. 7. 



The Permanence of the Practice Effect 



75 



TABLE XXVI 



(a) 
Gain in Accuracy in Addi- 
tion FROM Initial Score 
TO THE End of June 



(b) 
Gain in Accuracy of Same 
Individuals During the 
Experiment 



Per cent gained 


Individuals 


Per cents 


Individuals 


Per cents 


in accuracy 










-55 to -51 






4 


1.7 


-50 to -46 


1 


.5 






-45 to -41 


3 


1.2 


1 


.5 


-40 to -36 


1 


.5 


1 


.5 


-35 to -31 


3 


1.2 


6 


2.4 


-30 to -26 


5 


1.9 


9 


3.6 


-25 to -21 


16 


6.1 


7 ' 


2.6 


-20 to -16 


20 


7.8 


20 


7.8 


-15 to -11 


22 


8.5 


25 


9.7 


-10 to - 6 


21 


8.2 


23 


9. 


- 5 to -1 


32 


12.4 


32 


12.4 


to 4 


36 


13.9 


42 


16.3 


5 to 9 


37 


14.4 


28 


10.9 


10 to 14 


20 


7.S 


22 


8.5 


15 to 19 


12 


4.7 


17 


6.6 


20 to 24 


16 


6.1 


8 


3.1 


25 to 29 


8 


3.1 


2 


.8 


30 to 34 


2 


.8 


2 


.8 


35 to 39 


1 


.5 


3 


1.2 


40 to 44 


1 


.5 


2 


.8 


45 to 49 






3 


1.2 


50 to 54 


1 


.5 






55 to 59 






1 


.5 


Total 


258 


100. 


258 


100. 




Median 


.2 


Median -.4 




25 Percer 


itile -12. 


25 Percentile -12.2 




75 Percer 


itile 9. 


75 Percentile 8.7 




P.E. 


10.5 


P.E. 10.5 



Division 

Gross Gain: Only two classes in the division experiment 
were given the retention test in June. The other classes did not 
finish the experiment until the closing days of the term. One o£ 
these classes finished the experiment June y, the other June lo. 
Both took the retention test June 20. Table XXVII gives under 
(a) the gross gain of these two classes in number of combina- 
tions worked correctly in the retention test on June 20 over the 
initial practice-period of the experiment, and under (b) the gross 
gain of the same individuals during the experiment. The median 
gain in the retention test over the initial period was 30.7 com- 



76 



Practice in the Case of School Children 



binations while the gain during the experiment was 28.5 com- 
binations. This means that these two classes in an interval of 
about 12 days not only did not lose any of the proficiency acquired 
during the intense practice of the experiment but actually gained 
a median of 2.2 combinations. 

Gain in Accuracy: The change in accuracy may be seen in 
Table XXVIII. From the initial period to the retention test, the 
median gain was 2.7 per cent, while the median gain during 
the experiment was 2 per cent. These figures show a median 
loss of .7 per cent in accuracy in the interval from the close 
of the experiment to the retention test. 

TABLE XXVII 



(a) 
Gross Gain of Two 
Classes in Division from 
Initial Test to End of 
June 



(b) 
Gross Gain of Same 
Classes During Experi- 
ment 



Combinations 


Individuals 


Per cents 


Individuals 


Per cents 


gained 










-4 to 


1 


1.2 


1 


1.2 


1 to 5 


1 


1.2 


2 


2.4 


6 to 10 


3 


3.6 


6 


7.1 


11 to 15 


2 


2.4 


3 


3.5 


16 to 20 


7 


8.4 


10 


12.9 


21 to 25 


16 


19.3 


13 


15.3 


26 to 30 


11 


13.3 


11 


12.9 


31 to 35 


15 


18.1 


13 


15.3 


36 to 40 


10 


12.1 


4 


4.7 


41 to 45 


6 


7.2 


9 


10.6 


46 to 50 


3 


3.6 


5 


5.9 


51 to 55 


2 


2.4 


1 


1.2 


56 to 60 




1.2 


1 


2.4 


61 to 65 




12 


1 


1.2 


66 to 70 






1 


1.2 


71 to 75 




1.2 


2 


2.4 


76 to 80 




1.2 






81 to 85 










86 to 90 




1.2 






91 to 95 




1.2 






96 to 100 










Total 


S3 


100 


83 

1 


100 




Median 


30.7 


Median 


28.5 




25 Percer 


itile 22.6 


25 Percer 


itile 19.7 




75 Percer 


itile 38.6 


75 Percer 


itile 39 . 6 




P.E. 


8. 


P.E. 


9.9 



The Permanence of the Practice Effect 



77 



Summary: For two classes in division, in an interval of 
about 12 days between the close of the experiment and the reten- 
tion test, the median gross gain was 2.2 combinations done cor- 
rectly, and the median loss in accuracy was .7 per cent. Here, 
as in addition, these classes not only did not recede from the 
high efficiency reached during the experiment when practice 
ceased, but they made a slight gain while using these same 
associations in the performance of regular school work. 

Both the addition experiment and the division experiment indi- 
cate that these third- and fourth-year children who had improved 
their proficiency in adding and in doing the division combina- 
tions 50 per cent or more by intense practice in a brief space of 
time, lost none of this recently acquired proficiency so long as 
they continued to exercise these same functions to the extent 
demanded by the performance of regular school work. 

TABLE XXVIII 



(a) 
Gross Gain in Accuracy 
OF Two Classes in Divi- 
sion FROM Initial Test 
TO End of June 



(b) 
Gross Gain of Same 
Classes During Experi- 
ment 



Per cent gained 
in accuracy 


Individuals 


Per cents 


Individuals 


Per cents 


-25 to -21 








1 


1.2 


-20 to -16 












-15 to -11 


1 




1.2 






-10 to -6 


4 




4.8 


2 


2.4 


-5 to -1 


21 




25.3 


17 


20. 


to 4 


31 




37.1 


35 


41.1 


5 to 9 


16 




19.3 


14 


16.5 


10 to 14 


4 




4.8 


8 


9.4 


15 to 19 


2 




2.4 


3 


3.5 


20 to 24 


3 




3.6 


1 


1.2 


25 to 29 


1 




1.2 


2 


2.4 


30 to 34 








1 


1.2 


45 to 49 








1 


1.2 


Total 


83 


100. 


85 


100. 




Median 


2.0 


Median 


2.7 




25 Percentile 


-2.8 


25 Percer 


itile -.4 




75 Percentile 


6.1 


75 Percer 


itile 7.6 




P.E. 


4.5 


P.E. 


4. 



78 Practice in the Case of School Children 

Permanence of Associations Through Summer Vacation 

Our next problem is to measure the change in ability in addi- 
tion and division which occurred during the summer vacation, 
a time when the associations concerned fell into almost complete 
disuse. This change will be shown by comparing the scores in 
the retention tests given at the beginning of September with the 
scores made at the end of June. 

Addition 

Gross Loss: Five classes containing 152 children that had 
taken the practice during the latter half of the school year 191 1- 
1912 and the retention test at the end of June 1912 were 
accessible to the author the following September. School opened 
September 9. These five classes were given the retention tests 
September 10 and 11 with one exception. This class had been 
promoted into three or four other schools and could not be 
assembled for the test until September 16. Many children did 
not return in September. Others, of course, had taken their 
places, but only the records of children who were present at 
the end of June and on the day of the retention test in September 
could be used. Table XXIX gives under (a) the gross gain of 
these five classes in the September retention tests over their 
standing in the retention tests in June. The median loss in 
columns added correctly was 7.5 columns. The lower 25 per 
cent of the group lost 14. i columns or more and the upper 25 
per cent of the group lost 2. columns or less. Only the upper 
15 per cent of the group could add as many columns correctly in 
15 minutes at the beginning of September as at the end of June. 

Loss Per Cent: The loss per cent used in this portion of 
the discussion was computed by finding what per cent the gross 
loss was of the number of problems worked correctly in the 
retention test at the end of June. If a child added 40 columns 
correctly in the retention test in June and 32 in September his 
gross loss was 8 columns and his loss per cent was 20, which 
would appear in the table as — 20. These per cents were dis- 
tributed as shown under (b) in Table XXIX. The median 
change was a loss of 17 per cent; that is, the group lost during 
vacation, as shown by the median, 17 per cent of the ability which 
it had at the end of June. The lower 25 per cent of the indi- 



The Permanence of the Practice Effect 



79 



viduals lost 32 per cent or more, and the upper 25 per cent lost 
6. per cent or less. 

Loss in Accuracy: Anyone to whom the method of com- 
puting the change in accuracy is not clear may find this discussed 
under Table I. In Table XXIX, under (c), the gross gains in 
accuracy are distributed. The median loss was 4.2 per cent. The 
upper 35 per cent of the class did not lose in accuracy. 

TABLE XXIX 

Gain op Five Classes in Addition from End of June to Beginning 
OF September 





(a) 






(b) 






^0 




G 


ROSS Gain 


Gain Per Cent 


Gain in Accuract 


Columns 


Indi- 


Per 


Gain 


Indi- 


Per 


Per cent 


Indi- 


Per 


Gained 


viduals 


cent 


Per cent 


viduals 


cent 


Gained 


viduals 


cent 


-47 to -44 


2 


1.3 


-65 to -61 


2 


1.3 


-55 to -51 


2 


1.3 


-43 to -40 


1 


.7 


-60 to -56 


5 


3.3 


-50 to -46 


1 


.7 


-39 to -36 


2 


1.3 


-55 to -51 






-45 to -41 


3 


2. 


-35 to -32 






-50 to -46 


7 


4.6 


-40 to -36 


3 


2. 


-31 to -28 


4 


2.6 


-45 to -41 


4 


2.6 


-35 to -31 


3 


2. 


-27 to -24 


7 


4.6 


-40 to -36 


10 


6.6 


-30 to -26 


2 


1.3 


-23 to -20 


5 


3.3 


-35 to -31 


15 


10. 


-25 to -21 


7 


4.6 


-19 to -16 


12 


7.9 


-30 to -26 


9 


5.9 


-20 to -16 


10 


6.6 


-15 to -12 


14 


9.2 


-25 to -21 


13 


8.6 


-15 to -11 


21 


13.8 


-11 to -8 


29 


19.1 


-20 to -16 


16 


10.5 


-10 to -6 


17 


11.2 


-7 to -4 


29 


19.1 


-15 to -11 


17 


11.2 


-5 to -1 


28 


18.4 


-3 to -0 


24 


15.8 


-10 to -6 


16 


10.5 


Oto 4 


26 


16.5 


Ito 4 


19 


12.5 


-5 to -1 


12 


7.9 


5 to 9 


16 


10.5 


5 to 8 


2 


1.3 


Oto 4 


16 


10.5 


10 to 14 


6 


3.9 


9 to 12 


2 


1.3 


5 to 9 


5 


3.3 


15 to 19 


2 


1.3 








10 to 14 


1 


.7 


20 to 24 


2 


1.3 








15 to 19 


2 


1.3 


25 to 29 












20 to 24 


1 


.7 


30 to 34 


3 


2. 








25 to 29 


1 


.7 








Total 


152 


100. 




152 


100. 




152 


100. 


Median 




-7.5 


Median 


-1 


7. 


Median - 


4.2 


25 Perce 


ntile - 


14.1 


25 Percen 


tile -3 


2. 


25 Percentile -1 


S.8 


75 Perce 


ntile 


-2. 


75 Percen 


tile - 


6. 


75 Percentile 


2.8 


P.E. 




6.1 


P.E. 


1 


3. 


P.E. 


3.3 



Division 

Seven of the classes that had taken the practice during the 
last half of the preceding school year were accessible in Septem- 
ber. These classes were given the retention test on September 
10 and II except one class which could not be reassembled until 
September 16. Since all of these classes finished their practice 



So 



Practice in the Case of School Children 



in June, and since only two of them could be given the retention 
test in June, the September retention score is compared with the 
final practice-period of the practice in the experiment. 

Gross Loss: The gross losses were found by taking the differ- 
ence between the number of combinations done correctly in the 
retention test in September and the number done correctly in 
the final practice-period of the experiment in June. These losses 
for the 221 individuals comprising these seven classes were 
distributed in Table XXX under (a). The median loss for the 
group was 18.3 combinations. About 90 per cent of the group 
lost during vacation. The upper 25 per cent of the group lost 
10 or fewer combinations. 

TABLE XXX 

Gain in Seven Classes in Division Combinations from the End 

OF June to September 



(a) 
Gross Gain 



(b) 
Gain Per Cent 



(c) 
Gain in Accuracy 



Combina- 
tions 
Gained 



-64 to -60 

-59 to -55 

-54 to -50 

-49 to -45 

-44 to -40 

-39 to -35 

-34 to -30 

-29 to -25 

-24 to -20 

-19 to -15 

-14 to -10 

-9 to -5 

-4 to 

1 to 5 

6 to 10 

11 to 15 



Total 



Indi- 
viduals 



5 

1 

3 

4 

7 

9 

22 

2S 

23 

35 

34 

16 

17 



Per 
cent 



221 



2.3 

.5 

1.4 

1.8 

3.2 

4.1 

10. 

12.7 

10.4 

15.8 

15.4 

7.2 

7.7 

3.7 

3.7 

.5 



100. 



Gain 
Per cent 



-70 to 
-65 to 
-60 to 
-55 to 
-50 to 
-45 to 
-40 to 
-35 to 
-30 to 
-25 to 
-20 to 
-15 to 
-10 to 
-5 to 
Oto 
5 to 
10 to 
15 to 
20 to 
25 to 
30 to 
35 to 
40 to 
45 to 



-66 

-61 

-56 

-51 

-46 

-41 

-36 

-31 

-26 

-21 

-16 

-11 

-6 

-1 

4 

9 

14 

19 

24 

29 

34 

39 

44 

49 



Indi 
viduals 



1 

2 

3 

2 

9 

13 

11 

18 

19 

36 

25 

25 

15 

14 

11 

3 

4 

5 

1 

1 



221 



Per 
cent 



.5 
1. 
1.4 

.9 

4.1 

5.9 

5. 

8.2 

8.6 

16.3 

11.3 

11.3 

6.8 

6.3 

5. 

1.4 

1.8 

2.3 

.5 

.5 



1. 



100. 



Per cent 
gained in 
accuracy 



-55 to -51 

-50 to -46 
-45 to -41 
-40 to -36 
-35 to -31 
-30 to -26 
-25 to -21 
-20 to -16 
-15 to -11 
-10 to -6 
-5 to 

Oto 

5 to 
10 to 
15 to 
20 to 
25 to 
30 to 



Indi- 
viduals 



1 

1 

2 

1 

3 

6 

8 

12 

19 

42 

67 

44 

11 

3 



221 



Per 

cent 



.5 

.5 

.9 

.5 

1.4 

2.7 

3.7 

5.4 

8 6 

19. 

30.3 

19.9 

5. 

1.4 



100. 



Median -18.3 

25 Tercentile -28.7 

75 Percentile -10.3 

P.E. 9.2 



Median -21 . 

25 Percentile -32. 

75 Percentile -10. 

P.E. 11. 



Median -4 . 3 

25 Percentile -10.2 

75 Percentile -. 1 

P.E. 5. 



The Permanence of the Practice Effect 8 1 

Loss Per Cent: The loss per cent was computed by finding 
what per cent the gross loss was of the number of combinations 
worked correctly in the final practice-period of the experiment. 
If a child did 75 combinations correctly in the final practice- 
period in June and 60 combinations in the retention test in 
September the gross loss was 15 combinations, or 20 per cent. 
These per cents are distributed in Table XXX under (b). The 
median loss was 21 per cent. 

Loss in Accuracy: The losses in accuracy are given in Table 
XXX under (c). The median loss was 4.3 per cent. 

Both in addition and division there was a great loss in ability 
during the summer vacation. The loss in division was greater 
than that in addition. This difference may have been due to 
the fact that younger children took part in the division work 
than in the addition or it may have been due to the fact that 
the children had been working in addition longer than in 
division.* Both the difference in age of the children and the 
fact that they had been working in addition longer than in 
division would afford conditions for causing the addition associa- 
tions to be more firmly set than the division combinations and 
hence more likely to persist. 

Permanence in Terms of Advance Over the Initial 
Practice-period of the Original Experiment 

The permanence of the improvement may also be measured 
by the superiority of the scores attained in September to the 
scores made at the beginning of the original experiment. In 
presenting these facts I shall present also the gain made from 
the initial to the final period of the experiment itself. The 
amount of permanence can then be viewed as the superiority of 
the September scores to the initial or their superiority to the 
final scores of the original experiment. 

Addition 

Gross Gain: The records of five classes of 148 children are 
available. One of these classes finished the experiment April 14, 
two May 4, one May 27, and one June 18. The questions to be 

*There was not only the excess of 15 minutes of practice in the orig- 
inal experiment, but also the extra June 15 minute retention test. Only 
two-sevenths of the classes had this June retention test. 



82 



Practice in the Case of School Children 



answered are, What gain on their initial ability did the pupils 
of these five classes make from the initial practice-period to the 
beginning of September? and, How much of their initial ability 
did they lose from the end of the experiment to the beginning 
of September ? The data for answering these questions are given 
in Table XXXI. The table shows that in September the indi- 
viduals in question were still 10.4 columns ahead of their standing 
at the beginning of the experiment but that from the close of 
the practice to the beginning of September they lost a median 
of 6.4 columns. 

Change in Accuracy: Similar data concerning accuracy are 
found in Table XXXII. The median loss in accuracy from 



TABLE XXXI 



(a) 
Gross Gain in Addition 
FROM Initial Practice- 
Period TO THE Reten- 
tion Test in September 



(b) 
Gross Gain op the Same 
Individuals During the 
Experiment 



Columns 
gained 


Individuals 


Per cents 


Individuals 


Per cents 


-19 to -12 


1 


.7 






-11 to -8 


1 


4.7 


2 


1.4 


-7 to -4 


9 


6.1 


1 


.7 


-3 to -0 


12 


8.1 


12 


8.1 


1 to 4 


18 


12.2 


19 


12.8 


5 to 8 


19 


12.8 


16 


10.8 


9 to 12 


17 


11.5 


10 


6.8 


13 to 16 


17 


11.5 


13 


8.8 


17 to 20 


13 


8.8 


16 


10.8 


21 to 24 


13 


8.8 


11 


7.4 


25 to 28 


6 


4.1 


15 


10.1 


29 to 32 


6 


4.1 


8 


5.4 


33 to 36 


4 


2.7 


5 


3.4 


37 to 40 


3 


2. 


7 


4.7 


41 to 44 






1 


.7 


45 to 48 






3 


2. 


49 to 52 






4 


2.7 


53 to 56 










57 to 60 


1 


.7 


2 


1.4 


61 to 64 


2 


1.4 


1 


.7 


65 to 68 






2 


1.4 


Total 


148 


100. 


148 


100. 





Median 


10.4 


Median 


16.8 




25 Percentile 


2. 


25 Percentile 


5.3 




75 Percentile 


19.9 


25 Percentile 


27.2 




P.E. 


8.8 


P.E. 


11. 



The Permanence of the Practice Effect 



83 



the initial period to the beginning of September was 4.1 per 
cent, while the loss in accuracy from the beginning to the end 
of the original practice was 1.5. 



TABLE XXXII 



(a) 
Gain in Accuracy in Addi- 
tion FROM Initial Prac- 
tice-Period TO THE Re- 
tention Test in Septem- 
ber 



(b) 
Gain of Same Individuals 
IN Accuracy During 
THE Experiment 



Per cent gained 


Individuals 


Per cents 


Individuals 


Per cents 


in accuracy 










-55 to -51 


2 


1.4 


2 


1.4 


-50 to -46 


2 


1.4 






-45 to -41 


2 


1.4 


1 


.7 


-40 to -36 


1 


.7 






-35 to -31 


2 


1.4 


3 


2. 


-30 to -26 


8 


5.4 


4. 


2.7 


-25 to -21 


6 


4. 


5 


3.4 


-20 to -16 


11 


7.4 


12 


8.1 


-15 to -11 


17 


11,5 


13 


8.8 


-10 to -6 


18 


12.2 


14 


9.5 


-5 to -1 


18 


12.2 


25 


16.9 


to 4 


19 


12.8 


22 


14.9 


5 to 9 


17 


11.5 


15 


10.1 


10 to 14 


11 


7.4 


13 


8.8 


15 to 19 


5 


3.4 


11 


7.4 


20 to 24 


3 


2. 


2 


1.4 


25 to 29 


1 


.7 


4 


2.7 


30 to 34 


2 


1.4 


1 


.7 


35 to 39 


1 


.7 






40 to 44 


1 


.7 






45 to 49 










50 to 54 


1 


.7 


1 


.7 


Total 


148 


100. 


148 


100. 




Median 


-4.1 


Median 


-1.5 




25 Percen 


tile -14.6 


25 Percen 


tile -11.7 




75 Percen 


tile 6. 


75 Percen 


tile 7.8 




P.E. 


10.3 


P.E. 


9.8 



Division 

Gross Gain: The gross gain in number of combinations 
worked correctly from the initial period of the experiment to the 
beginning of September, and the gross gain of the same indi- 
viduals during the original experiment, are shown together in 



84 



Practice in the Case of School Children 



Table XXXIII. The median gain in the test at the beginning 
of September over the initial period was 17.5 combinations, while 
the median gain during the experiment was 37.5 combinations. 
At the beginning of September the group was thus still a median 
of 17.5 combinations ahead of the median ability shown in the 
initial period, but had lost a median of 20.0 combinations from 
the end of practice to the beginning of September. 



TABLE XXXIII 



(a) 
Gross Gain in Division 
FROM Initial Practice 
Period to the Reten- 
tion Test in September 



(b) 
Gross Gain of the Same 
Individuals During the 

Experiment 



ColuTiins 


Intli\'idual.s 


Per cents 


Individuals 


Per cents 


gained 










-14 to -10 


3 


1.4 


1. 


.5 


-9 to -5 


3 


1.4 


1 


.5 


-5 to 


12 


5.7 


2 


.9 


1 to 5 


28 


13.3 


2 


.9 


6 to 10 


16 


7.6 


9 


4.3 


11 to 15 


32 


15.2 


9 


4.3 


16 to 20 


28 


13.3 


17 


S. 


21 to 25 


16 


7.6 


19 


9. 


26 to 30 


21 


10 


18 


8.6 


31 to 35 


17 


8. 


20 


9.5 


36 to 40 


11 


5.2 


17 


8. 


41 to 45 


8 


3.9 


18 


8.6 


46 to 50 


8 


3.9 


14 


6.6 


51 to 55 


2 


.9 


12 


5.7 


56 to 60 


3 


1.5 


17 


8. 


61 to 65 


1 


.5 


10 


4.8 


66 to 70 






9 


4.3 


71 to 75 


1 


.5 


5 


2.4 


76 to 80 






2 


.9 


81 to 85 






4 


1.9 


86 to 90 






2 


.9 


91 to 95 






1 


.5 


96 to 100 






1 


.5 


Total 


210 


100. 


210 


100. 




Median 


17.5 


Median 


37.5 




25 Percer 


itile 7.5 


25 Percer 


itile 23.5 




75 Percer 


itile 30.1 


75 Percer 


tile 54.9 




P.E. 


11.3 


P.E. 


15.7 



Loss in Accuracy: Table XXXIV shows the facts. The 
median loss in accuracy from the beginning of the experiment to 



The Permanence of the Practice Effect 



85 



the beginning of September was .3 per cent while the median gain 
in accuracy during the experiment was 2.7 per cent. Hence 
during vacation there was a median loss of 3.0 per cent in 
accuracy. 

It must be remembered that these measures are from one 
15-minute test to another, and so do not measure tlie effect of 
the disuse of the associations pure and simple. The loss from 
the last half-minute of the original practice to the first half- 
minute of the test in September might well be greater. But for 
practical purposes we wish to know just the fact which is 
measured here; namely, the change in the ability as shown in a 
test of ordinary length. Whatever loss is recovered in a minute 
or two of practice is educationally of no moment. 

TABLE XXXIV 



(a) 
Gain in Accuracy from 
Initial Practice-Period 
TO THE Retention Iest 
in September 



(b) 
Gain op the Same Indi- 
viduals IN Accuracy 
During Experiment 



Per cent 
gained 


Individuals 


Per cents 


Individuals 


Per cents 


-40 to -36 






1 


.5 


-35 to -31 










-30 to -26 


2 


.9 


1 


5 


-25 to -21 


4 


1.9 






-20 to -16 


7 


3.3 






-15 to -11 


9 


4.3 


4 


1.9 


-10 to -6 


19 


9. 


7 


3.3 


-5 to -1 


61 


29. 


40 


19. 


to 4 


67 


31.9 


82 


39 


5 to 9 


20 


9.6 


39 


18.5 


10 to 14 


6 


2.9 


17 


8 


15 to 19 


10 


4.8 


7 


3.3 


20 to 24 


2 


.9 


4 


1.9 


25 to 29 






3 


1.4 


30 to 34 


1 


.5 


2 


.9 


35 to 39 


2 


.9 






40 to 44 






2 


.9 


45 to 49 






1 


.5 


Total 


210 


100. 


210 


100. 





Median 


-.3 


Median 


2.7 




25 Percentile 


-4.6 


25 Percentile 


-.5 




75 Percentile 


3.6 


75 Percentile 


7.4 




P.E. 


4.1 


P.E. 


4. 



86 Practice in the Case of School Children 

Permanence of Association in Addition and Division 
During Vacation as Shown by the Amount of Practice 
Required to Restore These Associations to Their Pre- 
vious Efficiency. 

As soon as possible after the retention tests in September, each 
class was given sufficient practice in addition and division to 
restore it approximately to the same efficiency it had attained 
at the end of its practice the preceding year. Here " approxi- 
mately " means that the standing of a class, as shown by an 
average that could be calculated quickly, was within a few 
units of its standing at the end of the practice experiment. 

This practice was conducted under the same conditions as the 
practice of the experiment except that a practice-period could 
not always be given to each class on successive school days since 
there were four classes in addition and six in division to be 
re-practiced. 

Addition 

Only four classes containing 114 of the children who had 
taken the practice the preceding year were given practice to 
make good the loss of vacation. 

Time of Practice: These four classes had their retention test 
and their practice as indicated below : 





Retention 




Final 


Class 


Practice Test 


Practice 


Practice 


IX 


Sept. 10—15 min. 


Sept. 16 — 5 min. 


Sept. 18-15 min. 


X 


Sept. 12—15 min. 




Sept. 17 — 15 min, 


XI 


Sept. 12—15 min. 


■ Sept. 17—5 min. 1 
■ Sept. 18 — 5 min. \ 


Sept. 17—15 min, 


VI 


Sept. 11 — 15 min. 


Sept. 25—15 min. 






. Sept. 19—15 min. J 





A wide difference in the amount of time required by different 
classes to regain their former efficiency is apparent. Class VI, 
which required the greatest amount of time, made a very great 
gain during the experiment of the previous year and so had a 
high standard to reach. This may partly account for the extra 
time required by this class to reach its former efficiency. 

The time required to regain this former efficiency was 
measured in the same way as was the time in the original experi- 
ment. While Class IX practiced 35 minutes, its gain was 
measured for only 273^ minutes. Its rate of adding during the 



The Permanence of the Practice Effect 87 

last 15-ininute period was the rate of the mid-point of that 
period, or at the end of 73^ minutes; hence the actual time 
required for this class to regain its former efficiency was the 
15 minutes of practice in the retention test of September 10, plus 
the 5 minutes on September 16, plus the 7^ minutes of the final 
practice-period, or 27^^ minutes. The time for the other classes 
was calculated in the same way. Accordingly Class X practiced 
225^2 minutes; Class XI, 22^ minutes; and Class VI, 473^ 
minutes. The average time required to regain approximately the 
efficiency shown at the end of the practice of the preceding year 
was 30 minutes for the four classes. In other words, 30 minutes 
of practice were required to bring this group approximately to 
the efficiency which it reached as a result of 60 minutes of 
practice the preceding year. 



COMPARATIVE STANDING OF THESE FOUR CLASSES AT THE END OF 
THE EXPERIMENT AND AT THE END OF SEPTEMBER PRACTICE 

Gross Gain: Table XXXV gives under (a) the gross gain 
of the pupils of these four classes at the end of practice in 
September over their standing in the initial period of the 
original experiment, and under (b) the gross gain of these same 
individuals during the original experiment. 

A comparison of these two gains will show how nearly the 
30 minutes of practice restored the group to its former efficiency. 
The median gain at the end of the 30 minutes of practice was 
12.5 columns, while the median gain for the same group during 
the experiment was 14.3 columns. This means that 30 minutes 
of practice restored the median ability of the group to within 
1.8 columns of the median ability of the group at the end of 
the experiment of the previous year. 

Accuracy: Table XXXVI shows that the median gain in 
accuracy for this group at the end of 30 minutes of practice over 
its accuracy in the initial practice-period was — 2 per cent. The 
loss of the group during the experiment was 3.1 per cent. 
Hence, so far as accuracy of performance is concerned, the 
condition of the group was practically the same at the end of 
the 30 minutes of practice in September as at the end of the 
experiment. 



88 



Practice in the Case of School Children 



The facts of Tables XXXV and XXXVI, in other words, 
are: 114 individuals who had taken part in the addition practice 
of the preceding year practiced again in September until they 
reached approximately the ability they had reached at the end 
of the experiment the preceding year. The average duration 
time of practice was 30 minutes, at the end of which time the 
group was within 1.8 columns of the standing which it had 
reached at the end of the 60 minutes of practice in the experi- 
ment of the previous year. The accuracy of performance was 
practically equal. Hence more than one-half as much practice 
was required in September to regain the standing reached in 
the experiment before vacation as was required at that time to 
reach it. 

TABLE XXXV 



(a) 
Gross Gain in Addition 
FROM THE Initial Prac- 
tice-Period OF the Ex- 
periment TO THE Last 
Practice in September 



(b) 
Gross Gain of the Same 
Individuals During the 
Experiment 



Columns 


Individuals 


Fer cents 


Individuals 


Per cents 


giiinetl 










-11 to - 8 


4 


3.5 


2 


1.7 


-7 to -4 


4 


3.5 


2 


1 7 


-3 to 


7 


6.1 


9 


7.9 


1 to 4 


14 


12.3 


16 


14. 


5 to 8 


[16 


14. 


13 


11.4 


9 to 12 


12 


10.5 


10 


8.7 


13 to 16 


14 


12.2 


11 


9.6 


17 to 20 


10 


8.7 


10 


8.7 


21 to 24 


5 


4.4 


8 


7 


25 to 28 


6 


5.3 


8 


7 


29 to 32 


7 


6.1 


4 


3 5 


33 to 36 


3 


2.6 


4 


3.5 


37 to 40 


3 


2 6 


6 


5.3 


41 to 44 






1 


.9 


45 to 48 


5 


4.4 


2 


1.7 


49 to 52 






3 


2.6 


53 to 56 


1 


.9 






57 to 60 


1 


.9 


2 


1.7 


61 to 64 


1 


.9 


1 


.9 


65 to 68 


1 


.9 


2 


1.7 


Total 


114 


100. 


114 


100. 




Median 


12.5 


Median 


14.3 




25 Percer 


tile 4.4 


25 Percen 


tile 4.4 




75 Percei 


tile 24.1 


75 Percei 


tile 26.8 




P.E. 


9.9 


P.E. 


10.2 



The Permanence of the Practice Effect 
TABLE XXXVI 



89 



(a) 
Gain in Accuracy in Addi- 
tion FROM THE Initial 
Practick-Pf:riod of the 
Experiment to the Last 
Period in September 



(b) 
Gain in Accuracy of the 
Same Individuals Dur- 
ing the Experiment 



Per cent 
gained 


Individuals 


Per cents 


Individuals 


Per cents 


-55 to -51 


1 


.9 


3 


2.0 


-50 to -46 


1 


.9 






-45 to -41 


2 


1.8 


1 


.9 


-40 to -36 


3 


2.6 






-35 to -31 


1 


.9 


4 


3.5 


-30 to -26 


5 


4.4 


4 


3.5 


-25 to -21 


3 


2.6 


5 


4.4 


-20 to -16 


8 


7 


10 


8.7 


-15 to -11 


10 


8.7 


11 


9.6 


-10 to -6 


16 


14 


10 


8.7 


-5 to -1 


10 


8.7 


19 


16.6 


to 4 


15 


12.1 


16 


14 


5 to 9 


10 


8.7 


10 


8.7 


10 to 14 


6 


5.3 


7 


6.1 


15 to 19 


9 


7.9 


7 


6.1 


20 to 24 


7 


0.1 


2 


1.8 


25 to 29 


4 


3.5 


3 


2.6 


30 to 34 


2 


1.7 


1 


.9 


35 to 39 


1 


.9 


1 


.9 


40 to 44 










Total 


114 


100. 


114 


100. 




Median 


-2. 


Median 


-3.1 




25 Percen 


itile -13.3 


25 Percen 


tile -14.8 




75 Percei] 


itile 9.9 


75 Percen 


tile 5.8 




P.E. 


11.6 


P.E. 


10.3 



Division 

Time of Practice: Only six classes, containing 163 children 
who had taken the practice the preceding year, could be reprac- 
ticed in September. These six classes had their retention test 
and their practice as follows : 



90 



Practice in the Case of School Children 



Retention 
Class Practice-test 

XII Sept. 10—10 min. 



X Sept, 10—10 min. 

XI Sept. 10—10 min. 

VII Sept. 10—10 min. 

VI Sept. 11—10 min, 

XIX Sept. 11—10 min. 



Practice 



Sept. 
Sept. 
Sept. 
Sept. 
Sept. 
Sept. 
Sept. 
Sept. 
Sept. 
Sept. 
Sept. 
sept. 



16— 5 m 
18—10 m 
16— 5 m 
18—10 m 
23—10 m 
16— 5 m 

16— 5 m 
18—10 m 

17 — 5 m 
18—10 m 
16— 5 m 
20—10 m 



n. [■ Sept. 25—10 min, 40 

n. J 

n 



Final Practice- 
period Amount 

Sept. 19—10 min. SO* 



1 

i 

n.J 



-I Sept. 18—10 min 

I Sor>+ 1Q in TTiin 



Sept. 19—10 min. 
Sept. 19—10 min. 
Sept. 25—10 min. 



20 
30 

30 

30 



The average time required for these classes to regain approxi- 
mately their former ability was 30 minutes.* In order to give 
an exact basis for comparing the standing of this group at the 
end of 30 minutes of practice in September with its standing 
after 50 minutes of practice in the experiment of the previous 
year, the facts of Tables XXXVII and XXXVIII are presented. 



COMPAIL\TIVE STANDING OF THESE SIX CLASSES AT THE END OF 
THE EXPERIMENT AND AT THE END OF PRACTICE IN SEPTEMBER 

Gross Gain: Table XXXVII shows that the median gain 
from the initial period of the original experiment to the end of 
the 30 minutes of practice in September was 30.3 combinations, 
while, during the original experiment itself, the median gain was 
34.2 combinations. The 30 minutes of practice lacked 3.9 com- 
binations of restoring the ability of this group to the ability it 
reached at the end of 50 minutes of practice the preceding spring. 

Accuracy: Table XXXVIII shows that the median gain in 
accuracy at the end of the 30 minutes of practice over the 
accuracy of the initial performance in the original experiment 
was 2.6 per cent; while during the experiment of the previous 
year a gain of 3.0 per cent was made. 

In other words Tables XXXVII and XXXVIII report as 
follows : 163 individuals who had taken part in the division 
practice of the preceding year practiced again in September 
until they reached approximately the ability reached at the end 

* Here, too, the rate for the last 10 minutes was the rate at the mid- 
point of the period or at the end of 5 minutes. Hence only 5 minutes of 
the last 10 minutes are included in the time required to restore the 
former ability. 



The Permanence of the Practice Effect 



91 



of the experiment the preceding year. The average duration of 
practice was 30 minutes, at the end of which time the group was 
within 3.9 combinations of the standing which it had reached 
at the end of the 50 minutes of practice in the experiment of 
the previous year. There was also a difference of .4 per cent 
in accuracy of performance. Hence more than three-fifths as 
much practice was required in September to regain the standing 
reached in the experiment of the previous year as was required 
at that time to reach it. 

TABLE XXXVII 



(a) 
Gross Gain in Division 
FROM THE Initial Period 
OF THE Experiment to 
THE Last Practice in 
September 



(b) 
Gross Gain of the Same 
Individuals During the 
Experiment 



Combinations 










gained 


Individuals 


Per cents 


Individuals 


Per cents 


-19 to -15 


1 


.6 






-14 to -10 


1 


.6 


1 


.6 


-9 to -5 


2 


1.2 






-4 to 


4 


2.5 


2 


1.2 


1 to 5 


5 


3. 


2 


1.2 


6 to 10 


8 


4.9 


8 


4.9 


11 to 15 


11 


6.8 


7 


4.3 


16 to 20 


16 


9.8 


17 


10.4 


21 to 25 


18 


11. 


19 


11.6 


26 to 30 


16 


9.8 


13 


8. 


31 to 35 


11 


6.8 


17 


10.4 


36 to 40 


15 


9.2 


14 


8.6 


41 to 45 


13 


8. 


15 


9.2 


46 to 50 


8 


4.9 


11 


6.8 


51 to 55 


10 


6.1 


9 


5.5 


56 to 60 


5 


3. 


11 


6.8 


61 to 65 


6 


3.7 


5 


3. 


66 to 70 


3 


1.9 


4 


2.5 


71 to 75 


2 


1.2 


4 


2 5 


76 to 80 


4 


2.5 






81 to 85 


1 


.6 


2 


1.2 


86 to 90 










91 to 95 


2 


1.2 


1 


.6 


96 to 100 


1 


.6 


1 


.6 


Total 


163 


100. 


163 


100. 




Median 


30.3 


Median 


34.2 




25 Percen 


tile 18.2 


25 Percen 


tile 21.5 




75 Percen 


tile 46.3 


75 Percen 


tile 48.8 




P.E. 


14. 


P.E. 


13.7 



92 



practice in the Case of School Children 
TABLE XXXVIII 



(a) 
Gain in Accuracy in Divi- 
sion FROM THE Initial 
Period of the Experi- 
ment TO THE Last Peri- 
od in September 



(b) 
Gain in Accuracy of the 
Same Individuals Dur- 
ing the Experiment 



Per cent 
gained 


Individuals 


Per cents 


Individuals 


Per cents 


-45 to -41 


1 


.6 






-40 to -35 


2 


1.2 


1 


.6 


-33 to -31 










-30 to -26 


1 


.6 


2 


1.2 


-25 to -21 










-20 to -16 


1 


.6 






-15 to- 11 


2 


1.2 


2 


1.2 


-10 to S 


10 


6.1 


6 


3.7 


-5 to -1 


30 


18.4 


30 


18.4 


to 4 


50 


34.3 


58 


35.5 


5 to 9 


26 


16. 


31 


19. 


10 to 14 


14 


8.6 


14 


8.6 


15 to 19 


9 


5.5 


7 


4.3 


20 to 24 


3 


1.9 


4 


2.5 


25 to 29 


4 


2.5 


3 


1.9 


30 to 34 


1 


.6 


2 


1.2 


35 to 39 


2 


1.2 






40 to 44 






2 


1.2 


45 to 49 


1 


.6 


1 


.6 


Total 


163 


100. 


163 


100. 




Median 


2.6 


Median 


3. 




25 Percer 


itile -1 . 5 


25 Percer 


Itile -.5 




76 Percer 


itile 8.2 


75 Percer 


Itile 8.2 




P.E. 


4.9 


P.E. 


4.4 



APPENDIX I 

The Use of the Method of the Practice Experiment in 
Teaching Handwriting and Spelling 

Handivriting: The method of the practice experiment was 
applied by the author to writing in five fourth-year classes. 
Pages of Robinson Crusoe were printed to serve as material for 
the classes to copy. Uniform penholders and pens were supplied 
in each class. The regular school paper, of good quality, white 
and ruled, was used for the writing. The children were supplied 
with more material than they could copy in the time allowed. 
They were told to write as rapidly and as well as they could. 
About equal stress was put upon quantity and quality. Proper 
position was encouraged. 

Class I practiced 3 minutes a day for 16 successive school days. 

Class II " 3 " "9 alternate " 

Class III " 1 " " 12 successive " " 

Class IV " 1 « "9 alternate 

Class V " 6 " " 12 successive " 

The papers were not scored during the practice so the children 
had no definite measure of the quantity and quality from one day 
to the next as they had in arithmetic. This omission made the 
method radically different, since the incentive from knowing 
their exact progress was lacking. However, the children were 
always asked if they had written more and better than on the 
previous day. The author and the grade teacher observed the 
writing and urged that improvement in quality was as desirable 
as improvement in quantity. 

The first and last days' papers were scored for the number of 
letters written. They were also scored for quality by six 
different judges, by means of Thorndike's Scale for Handwrit- 
ing, and the average judgment for individuals and class was 
found. There was a decided gain in quantity in all the classes 
and a gain in quality in two classes. The following table gives 
(i) the average number of letters written per minute in the 

93 



94 



Practice in the Case of School Children 



initial performance, (2) the average gain per minute in number 
of letters in the last day's performance over the first, and (3) 
the average change of quality expressed in steps of Thorndike's 
scale. 





Average Number of 

Letters Written per 

Minute in Initial 

Performance 


Average Number 

OP Letters 

Gained per 

Minute 


Average Change 

IN Steps op 

Thorndike's 

Scale 


Class I 


35 


7 


+ .02 


Class II 


24 


22 


—1.16 


Class III 


22 


32 


+ .17 


Class IV 


43 


21 


—1.53 


Class V 


17 


16 


— .9 



It is easy to interpret the results where there was a gain both 
in quantity and in quality as was the case in Classes I and III, 
the first of which gained 20 per cent in quantity and slightly 
in quality, the second of which gained 145 per cent in quantity 
and slightly in quality. Just what merit attaches to the per- 
formance where there was a very great gain in quantity, but a 
loss in quality, remains for future interpretation. 

The results suggest that it is possible by means of the practice 
experiment, to increase speed in writing in the case of fourth- 
year school children very much and at the same time increase 
the quality. Had there been a constant measure of quality to act 
as a check upon quantity, no doubt the classes that lost in quality 
might have transferred part of the effort from speed to quality, 
thus maintaining or improving the quality and at the same time 
increasing their speed. 

An interesting problem for future experiment would be one 
similar to that outlined above with the added feature of having 
copies of Thorndike's scale hanging in the room readily 
accessible to children by which they might measure at any time 
the quality of their own writing. This would serve as an incen- 
tive to improvement in quality along with speed. 

Spelling: Five fourth-year classes were given work in 
spelling with the practice experiment as a method. A list of 
words was used, each one of which 60 per cent of the fourth-year 
children in a large school of high grade had missed in a certain 
test. There were about 100 words in the list. These words were 
placed on cards, one on each card, which were shuffled until 



Appendix 95 

they were promiscuously arranged. Then these words were 
dealt out in three piles until 30 words were placed in each pile. 
These groups of 30 words were then recorded for use. Then 
these 90 words were shuffled as before. Next they were dealt 
out in six piles until there were 12 words in each pile. These 
lists were recorded for use. Then these 90 words were shuffled 
again as before. This time they were dealt out in fifteen piles 
of 6 words each. These lists of 6 words were recorded. 

The groups of 6 words and 12 words were printed on large 
cards to be hung in the front of the room during the study period. 
It was found impracticable to put the list of 30 words on such a 
card, so they were typewritten, a copy for each child. 

Class I had 15 minutes to study 30 words on each of 3 successive 
school days 

Class II had 6 minutes to study 12 words on each of 6 successive 
school days 

Class III had 6 minutes to study 12 words on each of 6 alternate 
school days 

Class IV had 3 minutes to study 6 words on each of 13 successive 
school days 

Class V had 3 minutes to study 6 words on each of 9 alternate 
school days 

The children were told that there were words on the cards 
which they should study when the cards were turned at a given 
signal. They were told how much time they would have for 
study. They were told to study in any way that they choose, 
except that they should not take up their pencils during the 
study. This was done to prevent any attempt at writing words 
so that they could be used later. They were given no further 
instructions as to how they should study, except that they should 
try to learn every word. 

Before the study was begun, papers and pencils were passed 
out to be ready for use immediately after the study ended. 

At a given signal the chart or typewritten paper was turned 
for study. They studied diligently until the chart was turned 
or the signal to turn their typewritten papers face down was 
given. Then the words on the chart were pronounced by the 
experimenter in a promiscuous order at the rate of about 4 words 
per minute, and written by the pupils. 

The papers were marked before the next day's practice. 
Before beginning the following day's study the scores were read, 



96 Practice in tJte Case of School Children 

and good records commended. All were urged to try to beat 

their previous day's record. This was continued for the number 

of days indicated above. 

The following is the record comprising (i") the percentage of 

words spelled correctly during the entire experiment. (2) the 

percentage of words spelled correctly in the tirst two tests, and 

(3) the percentage of words spelled correctly in the last two 

tests. 

Per Cknt of Wokds Per Cent of Words Per Cent of Words 

Sfelled Correctly Spelled Correctly Spelled Correctly 

IX Entire List in First IVo Tests in L.\st Two Tests 

Class I S2 84 S2 

Chiss II 91 87 92 

Class III S7 81 93 

Class IV 77 88 70 

Class V 93 95 90 

The results indicate, as have other studies of spelling by Corn- 
man and Rice, that spelling is not amenable to method. Class II 
and Class III showed improvement in the last two tests over 
the first two. but the other three classes showed a loss. 



APPENDIX II 

Samit-ic ok AnniTioN Siii:i:is 
Reduced to ;<i of the orij^inal size 



7 


^ 


9 


2 


3 


7 


2 


7 


8 


8 


3 


5 


6 


9 


8 


7 


8 


2 


3 


9 


7 


^ 


6 


5 


6 


6 


7 


9 


2 


4 


6 


3 


6 


3 


6 


6 


8 


8 


3 


2 


3 


3 


3 


5 


7 


7 


3 


8 


5 


2 


6 


3 


5 


3 


8 


7 


6 


8 


8 


6 


5 


8 


6 


5 


6 


9 


5 


7 


3 


3 


3 


5 


9 


«+ 


2 


8 


2 


8 


9 


3 


5 


5 


8 


2 


2 


7 


9 


8 


5 


5 


6 


2 


9 


2 


5 


6 


3 


8 


4 


9 


8 


5 


9 


6 


2 


7 


9 


9 


8 


9 


2 


6 


5 


7 


8 


6 


3 


7 


5 


H 


9 


4 


3 


7 


^ 


2 


9 


7 


7 


5 


3 


3 


9 


8 


2 


8 


7 


9 


6 


7 


2 


7 


7 


6 


7 


5 


2 


8 


8 


5 


5 


3 


2 


3 


8 


3 


7 


6 


2 


9 



896^736245^44567 
7476485579786729 
5465658654433996 
8734977898595264 



9 


6 


9 


9 


7 


9 


7 


3 


6 


2 


7 


6 


8 


4 


5 


9 


2 


5 


6 


2 


8 


4 


4 


8 


8 


9 


6 


2 


3 


8 


6 


7 


6 


9 


8 


7 


3 


4 


9 


8 


5 


6 


6 


4 


5 


3 


3 


5 


9 


4 


2 


2 


4 


8 


8 


7 


8 


3 


8 


9 


6 


9 


5 


9 


4 


4 


4 


9 


5 


9 


3 


3 


3 


8 


9 


2 


2 


8 


9 


8 


-3 


-8 


J_ 


_5 


4 


-§ 


_6 


_8 


_2 


_4 


_4 


_8 


_6 


_2 


_7 


_6 


7 


7 


5 


9 


2 


2 


3 


8 


8 


5 


6 


2 


3 


9 


6 


7 


5 


5 


8 


9 


7 


5 


5 


2 


4 


7 


6 


7 


9 


3 


7 


5 


8 


2 


3 


8 


9 


8 


7 


8 


2 


6 


7 


9 


8 


3 


9 


9 


7 


7 


6 


6 


4 


8 


2 


9 


5 


8 


4 


7 


7 


8 


5 


2 


2 


5 


4 


9 


4 


7 


6 


4 


9 


6 


4 


2 


3 


4 


7 


5 


6 


8 


7 


8 


6 


7 


8 


6 


7 


3 


5 


9 


2 


2 


6 


3 


9 


6 


6 


6 


6 


2 


7 


2 


5 


3 


4 


6 


3 


6 


4 


4 


2 


4 


5 


3 


3 


9 


9 


7 


9 


7 


7 


3 


4 


9 


5 


6 


6 


8 


9 


2 


5 


6 


4 


9 


2 


5 


9 


5 


7 


5 


4 


9 



-§-3-^_i_8._9_9_5_3._4_6^_9_6_9_9 

97 



9 8 Practice in the Case of School Children 

Sample of Division Sheets 
Reduced to about ^3 of original size 



20= 


5S* 




31 = 


> 
7s and 


r. 


22= 


6s and 




56= 


9s and 




83= 


93 and 


r. 


53= 


6s and 




30= 


7s and 




21 = 


7s 




33= 


4s and 




89= 


9s and 




54- 


8? and 


r. 


77= 


8s and 




20= 


8s and 




32= 


4s 




22= 


9s and 




56= 


6s and 




80= 


9s and 


r. 


52= 


7s and 




31 = 


4s and 




22= 


3s and 


r. 


33= 


7s and 




86= 


9s and 




53= 


9s and 


r. 


75= 


9s and 




21 = 


4s and 




32= 


7s and 


r. 


23= 


5s and 




55= 


7s and 




78= 


9s and 


r. 


51 = 


8s and 




34= 


4s and 




24= 


7s and 


r. 


36= 


7s and 




74= 


8s and 




50= 


6s and 


r. 


67= 


9s and 




23= 


8s and 




35= 


7s 




25= 


9s and 




45= 


9s 




69= 


9s and 


r. 


48= 


6s 




34= 


7s and 


r. 


25= 


3s and 


r. 


38= 


7s and 




72= 


9s' 




49= 


4 
7s 




66= 


9s and 




24= 


4s 




36= 


4s 




26= 


5s and 




50= 


9s and 




68= 


9s and 




47= 


8s and 




35= 


4s and 




25= 


6s and 




39= 


4s and 




71 = 


8s and 




48= 


9s and 




65= 


9s and 




26= 


8s and 




62= 


9s and 




38= 


4s and 




47= 


5s and 




28= 


3s and 




59= 


9s and 




39= 


7s and 




44= 


9s and 




29= 


5s and 




64= 


9s and 




40= 


5s 




42= 


7s 




27= 


4s and 




61 = 


9s and 




41 = 


8s and 




46= 


7s and 




28= 


6s and 




59= 


6s and 




37= 


4s and 




44= 


6s and 




29= 


8s and 




63= 


9s 




41 = 


5s and 




43= 


5s and 




27= 


7s and 




60= 


9s and 




57= 


8s and 




45= 


7s and 




28= 


9s and 




58= 


7s and 




37= 


7s and 




43= 


8s and 


!• 


31= 


4s and 




















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